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I wonder if there is a way to count the number of zero digit in an integer number by using only these operations: +, -, * and /
Others operations such as cast, div, mod, ... are not allowed.
Input: 16085021
Output: 2
It is a major restriction that numbers cannot be compared in a way to know that one is less than the other, and only equality checks can be done.
In my opinion that means there is nothing much else you can do than repeatedly add 1 to a variable until it hits a target (n) and derive from that what the least significant digit is of the original number n. This is of course terribly slow and not scalable, but it works in theory.
Here is a demo in Python using only ==, +, - and /:
n = 16085021
count = 0
while True:
# find out what least significant digit is
digit = 0
i = 0
while True:
if n == i:
break
digit = digit + 1
if digit == 10:
digit = 0
i = i + 1
# count any zero digit
if digit == 0:
count = count + 1
# shift that digit out of n
n = (n - digit) / 10
if n == 0:
break
print(count) # 2
Modulo can be implemented with subtractions: a % b = subtract b from a until you end up with something < b and that's the result. You say we can only use the == comparison operator, but we are only interested in modulo 10, so we can just check equality to 0, 1, ..., 9.
def modulo10WithSubtractions(a):
if a == 0 || a == 1 || ... || a == 9:
return a
while True:
a = a - b
if a == 0 || a == 1 || ... || a == 9:
return a
Integer division by 10 can be implemented in a similar fashion: a // 10 = add 10 to itself as many times as possible without exceeding a.
def div10WithAdditions(a):
if a == 0 || a == 1 || ... || a == 9:
return 0
k = 0
while True:
k += 1
if a - k*10 == 0 || a - k*10 == 1 || ... || a - k*10 == 9:
return k
Then we can basically do the classical algorithm:
count = 0
while True:
lastDigit = modulo10WithSubtractions(a)
if lastDigit == 0:
count += 1
a = div10WithAdditions(a)
if a == 0:
break
Asuuming / means integer division, then this snippet does the job.
int zeros = 0;
while(num > 0){
if(num / 10 == (num + 9) / 10){
zeros++;
}
num /= 10;
}
Problem statement:
A company sells bowls of various integer sized diameters (inches) and often customers buy a number of these bowls at once.
The company would like to reduce shipping costs by sending the minimum number of packages for an order of bowls to a given customer by finding an optimal nesting of the bowls.
The company has also decided to restrict the nestings with the following limitations:
No more than 3 bowls should be nested in one nesting.
A bowl can be nested inside another if it's smaller but not more than 3 inches smaller than the bowl it's directly nested within.
For example, a customer orders the following bowl sizes:
One 5" bowl
One 8" bowl
Two 11" bowls
One 12" bowl
Two 15" bowls
The follow is a possible (and optimal) nesting:
[15] [15,12,11] [11,8,5]
Is there an algorithm to always provide an optimal nesting?
I've looked through many similar questions here on stackoverflow and googled around, but can't find this exact problem, nor am I able to map any similar problems over to this problem space in a way that solves the problem.
This was actually posted in another forum by a real business owner. A number of the developers tried to help, ultimately finding a heuristic solution that provided an optimal solution most of the time but not always.
I can share the chosen algorithm one of the developers put forward as well as a few approaches I tried myself.
I'm just very curious about this problem and if there is an algorithm that can actually do this, or the best solution will be heuristic. If you can either give an idea of how to approach this, share an algorithm, or send a link to a similar problem that can be mapped to this one, that would be awesome.
This can be solved with dynamic programming in polynomial time.
The idea is that we ONLY care about how many boxes there are total, and how many boxes there are of different top bowl sizes. We don't care about the details beyond that. This is a polynomial amount of state, and so we can track through the calculation and enumerate one arrangement per possible state in a polynomial time. We then reconstruct the minimal packing of bowls into boxes from that arrangement.
class Arrangement:
def __init__(self, next_bowl, prev_arrangement=None):
self.prev_arrangement = prev_arrangement
self.add_rule = None
self.open1 = {}
self.open2 = {}
self.next_bowl = next_bowl
if prev_arrangement is None:
self.boxes = 0
for i in range(next_bowl, next_bowl + 4):
self.open1[i] = 0
self.open2[i] = 0
else:
self.boxes = prev_arrangement.boxes
for i in range(next_bowl, next_bowl + 4):
self.open1[i] = prev_arrangement.open1.get(i, 0)
self.open2[i] = prev_arrangement.open2.get(i, 0)
# This will be tuples of tuples.
def state(self):
open1 = (self.open1[i+self.next_bowl] for i in range(4))
open2 = (self.open2[i+self.next_bowl] for i in range(4))
return (open1, open2)
def next_arrangements(self, bowl):
base_arrangement = Arrangement(bowl, self)
base_arrangement.boxes += 1
base_arrangement.add_rule = ("new",)
old_count = self.open2.get(bowl, 0)
base_arrangement.open2[bowl] = old_count + 1
yield base_arrangement
for i in range(1, 4):
if 0 < self.open1.get(bowl+i, 0):
next_arrangement = Arrangement(bowl, self)
next_arrangement.open1[bowl+i] -= 1
next_arrangement.add_rule = ("open", 1, bowl+i)
yield next_arrangement
if 0 < self.open2.get(bowl+i, 0):
next_arrangement = Arrangement(bowl, self)
next_arrangement.open2[bowl+i] -= 1
next_arrangement.open1[bowl] += 1
next_arrangement.add_rule = ("open", 2, bowl+i)
yield next_arrangement
def find_boxes(self):
items = self._find_boxes()
boxes = items["full"]
for more_boxes in items["open1"].values():
boxes.extend(more_boxes)
for more_boxes in items["open2"].values():
boxes.extend(more_boxes)
return list(reversed(sorted(boxes)))
def _find_boxes(self):
if self.prev_arrangement is None:
return {
"full": [],
"open1": {},
"open2": {},
}
else:
items = self.prev_arrangement._find_boxes()
rule = self.add_rule
if rule[0] == "new":
if self.next_bowl not in items["open2"]:
items["open2"][self.next_bowl] = [[self.next_bowl]]
else:
items["open2"][self.next_bowl].append([self.next_bowl])
elif rule[0] == "open":
if rule[1] == 1:
box = items["open1"][rule[2]].pop()
box.append(self.next_bowl)
items["full"].append(box)
elif rule[1] == 2:
box = items["open2"][rule[2]].pop()
box.append(self.next_bowl)
if self.next_bowl not in items["open1"]:
items["open1"][self.next_bowl] = [box]
else:
items["open1"][self.next_bowl].append(box)
return items
def __str__ (self):
return str(self.boxes) + " open1:" + str(self.open1) + " open2:" + str(self.open2)
def bowl_nesting (bowls):
bowls = list(reversed(sorted(bowls))) # Largest to smallest.
start_arrangement = Arrangement(bowls[0])
arrange = {start_arrangement.state(): start_arrangement}
for bowl in bowls:
next_arrange = {}
for state, arrangement in arrange.items():
for next_arrangement in arrangement.next_arrangements(bowl):
state = next_arrangement.state()
if state in next_arrange and next_arrange[state].boxes <= next_arrangement.boxes:
pass # We are not an improvement.
else:
next_arrange[state] = next_arrangement
arrange = next_arrange
min_boxes = len(bowls)
min_box_list = None
for arrangement in arrange.values():
if arrangement.boxes <= min_boxes:
min_boxes = arrangement.boxes
min_box_list = arrangement.find_boxes()
return min_box_list
print(bowl_nesting([15, 15, 12, 11, 11,8,5]))
Now while the above solution works, it is inefficient. Suppose that we have up to k bowls of any given size. The number of combinations of open1[bowl] and open2[bowl] that allows is k choose 2 = k*(k-1)/2). When we consider that our state has 4 sizes in it, that's O(k^8 / 16 possible states. We do that for the number of bowls to get O(n k^8). This doesn't scale well.
We can do better by making the following notes:
In any arrangement with an open2[bowls+3] option, you do not do worse by moving the next bowl out of whatever box you were going to put it in, and putting it there instead.
If there is an open2[bowls+2] option and an open2[bowls+1] option, you never do worse by picking open2[bowls+2].
If there is an open1[bowls+i] option and an open1[bowls+j] option with 1 <= i < j <= 3 then you never do worse picking open1[bowls+i] instead.
This optimization means fewer choices, which speeds you up by a constant. But also you cannot have open2[bowls+3] and also have open2[bowls]. So that O(k^8) becomes O(k^7) states. And adding to the boxes with larger bowls will reduce how much of the potential state space we actually visit. This should lead to a better constant.
Here is this logic with a minor refactor to cleanup the code.
class Arrangement:
def __init__(self, next_bowl, prev_arrangement=None, choice=None, position=None):
self.prev_arrangement = prev_arrangement
self.add_rule = None
self.open1 = {}
self.open2 = {}
self.next_bowl = next_bowl
if prev_arrangement is None:
self.boxes = 0
for i in range(next_bowl, next_bowl + 4):
self.open1[i] = 0
self.open2[i] = 0
else:
self.boxes = prev_arrangement.boxes
for i in range(next_bowl, next_bowl + 4):
self.open1[i] = prev_arrangement.open1.get(i, 0)
self.open2[i] = prev_arrangement.open2.get(i, 0)
if choice is not None:
self.choice(choice, position)
# This will be tuples of tuples.
def state(self):
open1 = (self.open1[i+self.next_bowl] for i in range(4))
open2 = (self.open2[i+self.next_bowl] for i in range(4))
return (open1, open2)
def choice (self, rule, position=None):
self.add_rule = (rule, position)
if rule == "new":
self.boxes += 1
self.open2[self.next_bowl] += 1
elif rule == "open1":
self.open1[position] -= 1
elif rule == "open2":
self.open2[position] -= 1
self.open1[self.next_bowl] += 1
def next_arrangements(self, bowl):
if 0 < self.open2.get(bowl+3, 0):
yield Arrangement(bowl, self, "open2", bowl+3)
else:
yield Arrangement(bowl, self, "new")
for i in [3, 2, 1]:
if 0 < self.open1.get(bowl+i, 0):
yield Arrangement(bowl, self, "open1", bowl+i)
break
for i in [2, 1]:
if 0 < self.open2.get(bowl+i, 0):
yield Arrangement(bowl, self, "open2", bowl+i)
break
def find_boxes(self):
items = self._find_boxes()
boxes = items["full"]
for more_boxes in items["open1"].values():
boxes.extend(more_boxes)
for more_boxes in items["open2"].values():
boxes.extend(more_boxes)
return list(reversed(sorted(boxes)))
def _find_boxes(self):
if self.prev_arrangement is None:
return {
"full": [],
"open1": {},
"open2": {},
}
else:
items = self.prev_arrangement._find_boxes()
rule = self.add_rule
if rule[0] == "new":
if self.next_bowl not in items["open2"]:
items["open2"][self.next_bowl] = [[self.next_bowl]]
else:
items["open2"][self.next_bowl].append([self.next_bowl])
elif rule[0] == "open1":
box = items["open1"][rule[1]].pop()
box.append(self.next_bowl)
items["full"].append(box)
elif rule[0] == "open2":
box = items["open2"][rule[1]].pop()
box.append(self.next_bowl)
if self.next_bowl not in items["open1"]:
items["open1"][self.next_bowl] = [box]
else:
items["open1"][self.next_bowl].append(box)
return items
def bowl_nesting (bowls):
bowls = list(reversed(sorted(bowls))) # Largest to smallest.
start_arrangement = Arrangement(bowls[0])
arrange = {start_arrangement.state(): start_arrangement}
for bowl in bowls:
next_arrange = {}
for state, arrangement in arrange.items():
for next_arrangement in arrangement.next_arrangements(bowl):
state = next_arrangement.state()
if state in next_arrange and next_arrange[state].boxes <= next_arrangement.boxes:
pass # We are not an improvement.
else:
next_arrange[next_arrangement.state()] = next_arrangement
arrange = next_arrange
min_boxes = len(bowls)
min_box_list = None
for arrangement in arrange.values():
if arrangement.boxes <= min_boxes:
min_boxes = arrangement.boxes
min_box_list = arrangement.find_boxes()
return min_box_list
print(bowl_nesting([15, 15, 12, 11, 11,8,5]))
Yes, we can calculate an optimal nesting. As you presented, start with the bowls sorted in reverse order.
15,15,12,11,11,8,5
Assign the minimum number of starting bowls, corresponding to the count of the largest bowl.
[15] [15]
As we iterate element by element, the state we need to keep is the smallest bowl size and count in each container per index visited.
index 0, [(15, 1), (15, 1)]
(The state can be further refined to a multiset of those packages with identical count and smallest bowl size, which would add some complication.)
The choice for any element is which box (or set of boxes with similar state) to add it to or whether to start a new box with it.
index 1, [(15, 1), (12, 2)]
or
index 1, [(15, 1), (15, 1), (12, 1)]
We can explore these branches in an iterative or recursive breadth first search prioritised by the number of elements remaining plus the number of packages in the state, avoiding previously seen states.
We can further prune the search space by avoiding branches with the same or more count of packages than the best we've already seen.
This approach would amount to brute force in the sense of exploring all relevant branches. But hopefully the significant restrictions of package size and bowl size relationship would narrow the search space considerably.
This "Answer" is based on btilly's solution (the accepted answer).
Thank you #btilly for sticking with this and taking the time to revise the algorithm and fix bugs!
Since this was originally set within the context of Google Apps Script, I've rewritten this in Javascript and want to share the JS code with anyone else that might want it.
btilly's improved algorithm does indeed run much quicker than the first. Though the improvement factor depends on the bowls provided I've noticed it running up to 50 times faster in some of my sample sets.
Below is the JS code. Some caveats:
I've kept the same structure and same naming as much as possible in copying over btilly's solution.
There's no guarantee I did not introduce bugs while porting over btilly's code.
I'm not too familiar with many modern/proper JS conventions and also I don't know Python at all, so translating some of the concepts was tough and although I think my code is now bug free, if you spot any bugs, inefficiencies, bad programming ideas, please let me know and I'll update the below code.
I added a count to the state creation to make each state unique, since in my Apps Script implementation the JS runtime kept stringifying the arrays so that two states were sometimes considered the same even if they were not (e.g. the previous arrangement's bowl was the same size as another arrangement's bowl, but not the same bowl - the way two 10" bowls might appear to a 9" bowl for example). This was not needed in Python since the generators were unique based on their memory addresses. If you know a better way to do this in JS, please let me know. Seems a little sloppy the way I did it.
Improved/faster code (Javascript):
class Arrangement2{
constructor(next_bowl, prev_arrangement, choice, position){
this.prev_arrangement = prev_arrangement;
this.add_rule = null;
this.open1 = {};
this.open2 = {};
this.next_bowl = next_bowl;
if (prev_arrangement == null){
this.boxes = 0;
for (let i = next_bowl; i < next_bowl + 4; i++){
this.open1[i] = 0;
this.open2[i] = 0;
}
}
else{
this.boxes = prev_arrangement.boxes;
for (let i = next_bowl; i < next_bowl + 4; i++){
this.open1[i] = prev_arrangement.open1[i] != null ? prev_arrangement.open1[i] : 0;
this.open2[i] = prev_arrangement.open2[i] != null ? prev_arrangement.open2[i] : 0;
}
}
if(choice != null){
this.choice(choice,position);
}
}
state(){
let open1 = {};
let open2 = {};
for(let i = 0; i < 4; i++){
open1[i+this.next_bowl] = this.open1[i+this.next_bowl];
open2[i+this.next_bowl] = this.open2[i+this.next_bowl];
}
var toReturn = [];
//Used to make each state unique, without this the algorithm may not always find the best solution
Arrangement2.count++;
toReturn.push(Arrangement2.count);
toReturn.push(open1);
toReturn.push(open2);
return toReturn;
}
choice(rule, position){
this.add_rule = [rule, position];
if( rule == "new" ){
this.boxes += 1;
this.open2[this.next_bowl] += 1;
}
else if( rule == "open1" ){
this.open1[position] -= 1;
}
else if( rule == "open2" ){
this.open2[position] -= 1;
this.open1[this.next_bowl] += 1;
}
}
* next_arrangements (bowl){
if( 0 < (this.open2[bowl+3] != null ? this.open2[bowl+3] : 0)){
yield new Arrangement2(bowl, this, "open2", bowl + 3);
}
else{
yield new Arrangement2(bowl, this, "new", null);
for(let i = 3; i > 0; i--){
if (this.open1[bowl+i] != null ? this.open1[bowl+i] : 0){
yield new Arrangement2(bowl, this, "open1", bowl+i);
break ;
}
}
for(let i = 2; i > 0; i--){
if (this.open2[bowl+i] != null ? this.open2[bowl+i] : 0){
yield new Arrangement2(bowl, this, "open2", bowl+i);
break ;
}
}
}
}
find_boxes(){
let items = this._find_boxes();
let boxes = items["full"];
for (const [key, more_boxes] of Object.entries(items["open1"])) {
boxes = boxes.concat(more_boxes);
}
for (const [key, more_boxes] of Object.entries(items["open2"])) {
boxes = boxes.concat(more_boxes);
}
//Max --> Min (i.e [ 12, 12, 11, 11, 10, 7, 7, 7 ])
boxes.sort(function(a, b){return b - a});
return boxes; //boxes.sort().reverse(); //list(reversed(sorted(boxes)));
}
_find_boxes(){
if (this.prev_arrangement == null){
return {
"full": [],
"open1": {},
"open2": {},
}
}
else{
let items = this.prev_arrangement._find_boxes();
let rule = this.add_rule;
if (rule[0] == "new"){
if (!(this.next_bowl in items["open2"])){
items["open2"][this.next_bowl] = [[this.next_bowl]];
}
else{
items["open2"][this.next_bowl].push([this.next_bowl]);
}
}
else if( rule[0] == "open1"){
let box = items["open1"][rule[1]].pop();
box.push(this.next_bowl);
items["full"].push(box);
}
else if( rule[0] == "open2"){
let box = items["open2"][rule[1]].pop();
box.push(this.next_bowl);
if (!(this.next_bowl in items["open1"])){
items["open1"][this.next_bowl] = [box];
}
else{
items["open1"][this.next_bowl].push(box);
}
}
return items;
}
}
__str__(){
return this.next_bowl + " " + JSON.stringify(this.boxes) + " open1:" + JSON.stringify(this.open1) + " open2:" + JSON.stringify(this.open2);
}
}
allStates_nesting_improved = function (bowls){
//Used to make each state unique, without this the algorithm may not always find the best solution
Arrangement2.count = 0;
//Max --> Min (i.e [ 12, 12, 11, 11, 10, 7, 7, 7 ])
bowls.sort(function(a, b){return b - a});
let start_arrangement = new Arrangement2(bowls[0], null);
let returnObj = start_arrangement.state();
let arrange = {[returnObj]:start_arrangement};
for (const [key, bowl] of Object.entries(bowls) ) {
let next_arrange = {};
for (let [state, arrangement] of Object.entries(arrange) ) {
let next_arrangements = arrangement.next_arrangements(bowl);
let next_arrangement = next_arrangements.next();
while(next_arrangement.value != undefined){
next_arrangement = next_arrangement.value;
let state = next_arrangement.state();
let nextArrange_state = next_arrange[state];
if ( next_arrange[state] != undefined && (nextArrange_state === state) && next_arrange[state].boxes <= next_arrangement.boxes){
continue ; // # We are not an improvement.
}
else{
next_arrange[next_arrangement.state()] = next_arrangement;
}
next_arrangement = next_arrangements.next();
}
}
arrange = next_arrange;
}
let min_boxes = bowls.length;
let min_box_list = null;
for (const [key, arrangement] of Object.entries(arrange) ) {
if (arrangement.boxes <= min_boxes){
min_boxes = arrangement.boxes;
min_box_list = arrangement.find_boxes();
}
}
console.log(min_box_list);
return min_box_list;
}
Original code (Javascript):
class Arrangement1{
constructor(next_bowl, prev_arrangement){
this.prev_arrangement = prev_arrangement;
this.add_rule = null;
this.open1 = {};
this.open2 = {};
this.next_bowl = next_bowl;
if (prev_arrangement == null){
this.boxes = 0;
for (let i = next_bowl; i < next_bowl + 4; i++){
this.open1[i] = 0;
this.open2[i] = 0;
}
}
else{
this.boxes = prev_arrangement.boxes;
for (let i = next_bowl; i < next_bowl + 4; i++){
this.open1[i] = prev_arrangement.open1[i] != null ? prev_arrangement.open1[i] : 0;
this.open2[i] = prev_arrangement.open2[i] != null ? prev_arrangement.open2[i] : 0;
}
}
}
state(){
//Used to make each state unique, without this the algorithm may not always find the best solution
Arrangement1.count++;
let open1 = {};
let open2 = {};
for(let i = 0; i < 4; i++){
open1[i+this.next_bowl] = this.open1[i+this.next_bowl];
open2[i+this.next_bowl] = this.open2[i+this.next_bowl];
}
var toReturn = [];
toReturn.push(Arrangement1.count);
toReturn.push(open1);
toReturn.push(open2);
return toReturn;
}
* next_arrangements (bowl){
let base_arrangement = new Arrangement1(bowl, this);
base_arrangement.boxes += 1;
base_arrangement.add_rule = ["new"];
let old_count = this.open2[bowl] != null ? this.open2[bowl] : 0;
base_arrangement.open2[bowl] = old_count + 1;
yield base_arrangement;
for(let i = 1; i < 4; i++){
if (0 < (this.open1[bowl+i] != null ? this.open1[bowl+i] : 0)){
let next_arrangement = new Arrangement1(bowl, this);
next_arrangement.open1[bowl+i] -= 1;
next_arrangement.add_rule = ["open", 1, bowl+i];
yield next_arrangement;
}
if (0 < (this.open2[bowl+i] != null ? this.open2[bowl+i] : 0)){
let next_arrangement = new Arrangement1(bowl, this);
next_arrangement.open2[bowl+i] -= 1;
next_arrangement.open1[bowl] += 1;
next_arrangement.add_rule = ["open", 2, bowl+i];
yield next_arrangement;
}
}
}
find_boxes(){
let items = this._find_boxes();
let boxes = items["full"];
for (const [key, more_boxes] of Object.entries(items["open1"])) {
boxes = boxes.concat(more_boxes);
}
for (const [key, more_boxes] of Object.entries(items["open2"])) {
boxes = boxes.concat(more_boxes);
}
//Max --> Min (i.e [ 12, 12, 11, 11, 10, 7, 7, 7 ])
boxes.sort(function(a, b){return b - a});
return boxes;
}
_find_boxes(){
if (this.prev_arrangement == null){
return {
"full": [],
"open1": {},
"open2": {},
}
}
else{
let items = this.prev_arrangement._find_boxes();
let rule = this.add_rule;
if (rule[0] == "new"){
if (!(this.next_bowl in items["open2"])){
items["open2"][this.next_bowl] = [[this.next_bowl]];
}
else{
items["open2"][this.next_bowl].push([this.next_bowl]);
}
}
else if( rule[0] == "open"){
if (rule[1] == 1){
let box = items["open1"][rule[2]].pop();
box.push(this.next_bowl);
items["full"].push(box);
}
else if( rule[1] == 2){
let box = items["open2"][rule[2]].pop();
box.push(this.next_bowl);
if (!(this.next_bowl in items["open1"])){
items["open1"][this.next_bowl] = [box];
}
else{
items["open1"][this.next_bowl].push(box);
}
}
}
return items;
}
}
__str__(){
return this.next_bowl + " " + JSON.stringify(this.boxes) + " open1:" + JSON.stringify(this.open1) + " open2:" + JSON.stringify(this.open2);
}
}
allStates_nesting = function (bowls){
//Used to make each state unique, without this the algorithm may not always find the best solution
Arrangement1.count = 0;
//Max --> Min (i.e [ 12, 12, 11, 11, 10, 7, 7, 7 ])
bowls.sort(function(a, b){return b - a});
let start_arrangement = new Arrangement1(bowls[0], null);
let returnObj = start_arrangement.state();
let arrange = {[returnObj]:start_arrangement};
for (const [key, bowl] of Object.entries(bowls) ) {
let next_arrange = {};
for (let [state, arrangement] of Object.entries(arrange) ) {
let next_arrangements = arrangement.next_arrangements(bowl);
let next_arrangement = next_arrangements.next();
while(next_arrangement.value != undefined){
next_arrangement = next_arrangement.value;
let state = next_arrangement.state();
let nextArrange_state = next_arrange[state];
if ( next_arrange[state] != undefined && (nextArrange_state === state) && next_arrange[state].boxes <= next_arrangement.boxes){
continue ; // # We are not an improvement.
}
else{
next_arrange[state] = next_arrangement;
}
next_arrangement = next_arrangements.next();
}
}
arrange = next_arrange;
}
let min_boxes = bowls.length;
let min_box_list = null;
for (const [key, arrangement] of Object.entries(arrange) ) {
if (arrangement.boxes <= min_boxes){
min_boxes = arrangement.boxes;
min_box_list = arrangement.find_boxes();
}
}
return min_box_list;
}
See it in action
Here is a link to a spreadsheet testbed with 3 algorithms:
Algorithm 1: A heuristic algorithm another developer provided (runs fast but doesn't always find the optimal solution and ignores some of the requirements in some of its solutions for simplicity's sake)
Algorithm 2: btilly's revised algorithm (faster)
Algorithm 3: btilly's first attempt
Bowl Nesting Spreadsheet
Feel free to make a copy and modify the code and/or add your own algorithm to compare it with the others. (The orange "Run" button won't work since the spreadsheet is in "Viewer" mode. You'll need to make a copy to run it).
To make a copy go to
File -> Make a copy.
Once you have your own copy, you can click the "Run" button or go to the code by clicking
Extensions -> Apps Script
You can then modify and/or add your own algorithm to the mix.
You'll also have to authorize the script to run as with all Apps Script scripts.
If you're worried about authorizing it, of course check out the code before clicking run to make sure there isn't anything nefarious in there.
I am trying to come up with a matching algorithm that matches on 3 attributes and I can't think of an effective solution. Here is the psuedocode for my algorithm
The algorithm does the following:
Check if there is a match on the first criteria.
If there is match on first criteria then I try to narrow the match down based on the other 2
criterias.
If there is no match on the first criteria, then try to
make a match on the second criteria.
If there is a match on the
second criteria, then I try to narrow the match down based on the
last criteria.
Etc...
It seems like this algorithm just repeats itself, and if I add another value to match on, then this algorithm grows very large real quickly.
//try to match on criteria 1
if results exist for match on criteria 1 {
//try to match on criteria 1 & 2
if results exist for match on criteria 1 & 2 {
//try to match on criteria 3
if result exist for match on criteria 1,2,3 {
return results for match on 1,2,3
}
else
return results for match on 1,2
}
else
return results for match on 1
}
//try to match on criteria 2
else if results exist for match on criteria 2 {
//try to match on criteria 2 & 3
if result exist for match on criteria 2,3 {
return results for match on 2,3
}
else
return results for match on 2
}
//try to match on criteria 3
else if results exist for match on criteria 3 {
return results for match on 3
}
else {
no match
}
Is there any better way to do this? It seems like
Here is an implementation in Javascript if I understand you correctly.
function filter(arr, condition) {
var retval = [];
for (i in arr) if (condition(arr[i])) retval.push(arr[i]);
return retval;
}
function first_match(arr, conditions) {
if (conditions.length == 0) return("No match");
var initial_arr = Array.prototype.slice.call(arr);
var prev_arr;
var i = 0;
for (; i < conditions.length && arr.length != 0; i++) {
prev_arr = Array.prototype.slice.call(arr);
arr = filter(arr, conditions[i]);
}
if (i <= 1 && arr.length == 0) return first_match(initial_arr, Array.prototype.slice.call(conditions, 1));
else if (i == conditions.length && arr.length != 0) return arr;
else return prev_arr;
}
Examples
var conditions = [function(x) { return x > 3; },
function(x) { return x % 2 == 0; },
function(x) { return Math.sqrt(x) % 1 == 0; }];
var example1 = [1,2,3,4,5,6,7,8,9,10];
console.log(first_match(example1, conditions)); // [4] - all three hold
var example2 = [1,2,3,5,6,7,8,9,10];
console.log(first_match(example2, conditions)); // [6, 8, 10] - First two hold
var example3 = [5, 7, 9];
console.log(first_match(example3, conditions)); // [5, 7, 9] - First one holds
var example4 = [-2, 0, 2];
console.log(first_match(example4, conditions)); // [0] - 2 & 3 hold
var example5 = [-2, 2];
console.log(first_match(example5, conditions)); // [-2, 2] - Only 2 holds
var example6 = [1];
console.log(first_match(example6, conditions)); // [1] - Last one holds
var example7 = [-1];
console.log(first_match(example7, conditions)); // "No match" - None hold
What is the most efficient way to convert numeric amount into English words
e.g. 12 to twelve
127 to one hundred twenty-seven
That didn't take long. This is an implementation written in Java.
http://snippets.dzone.com/posts/show/3685
Code
public class IntToEnglish {
static String[] to_19 = { "zero", "one", "two", "three", "four", "five", "six",
"seven", "eight", "nine", "ten", "eleven", "twelve", "thirteen",
"fourteen", "fifteen", "sixteen", "seventeen", "eighteen", "nineteen" };
static String[] tens = { "twenty", "thirty", "forty", "fifty", "sixty", "seventy", "eighty", "ninety"};
static String[] denom = { "",
"thousand", "million", "billion", "trillion", "quadrillion",
"quintillion", "sextillion", "septillion", "octillion", "nonillion",
"decillion", "undecillion", "duodecillion", "tredecillion", "quattuordecillion",
"sexdecillion", "septendecillion", "octodecillion", "novemdecillion", "vigintillion" };
public static void main(String[] argv) throws Exception {
int tstValue = Integer.parseInt(argv[0]);
IntToEnglish itoe = new IntToEnglish();
System.out.println(itoe.english_number(tstValue));
/* for (int i = 0; i < 2147483647; i++) {
System.out.println(itoe.english_number(i));
} */
}
// convert a value < 100 to English.
private String convert_nn(int val) throws Exception {
if (val < 20)
return to_19[val];
for (int v = 0; v < tens.length; v++) {
String dcap = tens[v];
int dval = 20 + 10 * v;
if (dval + 10 > val) {
if ((val % 10) != 0)
return dcap + "-" + to_19[val % 10];
return dcap;
}
}
throw new Exception("Should never get here, less than 100 failure");
}
// convert a value < 1000 to english, special cased because it is the level that kicks
// off the < 100 special case. The rest are more general. This also allows you to
// get strings in the form of "forty-five hundred" if called directly.
private String convert_nnn(int val) throws Exception {
String word = "";
int rem = val / 100;
int mod = val % 100;
if (rem > 0) {
word = to_19[rem] + " hundred";
if (mod > 0) {
word = word + " ";
}
}
if (mod > 0) {
word = word + convert_nn(mod);
}
return word;
}
public String english_number(int val) throws Exception {
if (val < 100) {
return convert_nn(val);
}
if (val < 1000) {
return convert_nnn(val);
}
for (int v = 0; v < denom.length; v++) {
int didx = v - 1;
int dval = new Double(Math.pow(1000, v)).intValue();
if (dval > val) {
int mod = new Double(Math.pow(1000, didx)).intValue();
int l = val / mod;
int r = val - (l * mod);
String ret = convert_nnn(l) + " " + denom[didx];
if (r > 0) {
ret = ret + ", " + english_number(r);
}
return ret;
}
}
throw new Exception("Should never get here, bottomed out in english_number");
}
}
One way to accomplish this would be to use look up tables. It would be somewhat brute force, but easy to setup. For instance, you could have the words for 0-99 in one table, then a table for tens, hundreds, thousands, etc. Some simple math will give you the index of the word you need from each table.
This is some old python code on my hard drive. There might be bugs but it should show the basic idea:
class Translator:
def transformInt(self, x):
translateNumbers(0,[str(x)])
def threeDigitsNumber(self,number):
snumber=self.twoDigitsNumber(number/100)
if number/100!=0:
snumber+=" hundred "
snumber+self.twoDigitsNumber(number)
return snumber+self.twoDigitsNumber(number)
def twoDigitsNumber(self,number):
snumber=""
if number%100==10:
snumber+="ten"
elif number%100==11:
snumber+="eleven"
elif number%100==12:
snumber+="twelve"
elif number%100==13:
snumber+="thirteen"
elif number%100==14:
snumber+="fourteen"
elif number%100==15:
snumber+="fifteen"
elif number%100==16:
snumber+="sixteen"
elif number%100==17:
snumber+="seventeen"
elif number%100==18:
snumber+="eighteen"
elif number%100==19:
snumber+="nineteen"
else:
if (number%100)/10==2:
snumber+="twenty-"
elif (number%100)/10==3:
snumber+="thirty-"
elif (number%100)/10==4:
snumber+="forty-"
elif (number%100)/10==5:
snumber+="fifty-"
elif (number%100)/10==6:
snumber+="sixty-"
elif (number%100)/10==7:
snumber+="seventy-"
elif (number%100)/10==8:
snumber+="eighty-"
elif (number%100)/10==9:
snumber+="ninety-"
if (number%10)==1:
snumber+="one"
elif (number%10)==2:
snumber+="two"
elif (number%10)==3:
snumber+="three"
elif (number%10)==4:
snumber+="four"
elif (number%10)==5:
snumber+="five"
elif (number%10)==6:
snumber+="six"
elif (number%10)==7:
snumber+="seven"
elif (number%10)==8:
snumber+="eight"
elif (number%10)==9:
snumber+="nine"
elif (number%10)==0:
if snumber!="":
if snumber[len(snumber)-1]=="-":
snumber=snumber[0:len(snumber)-1]
return snumber
def translateNumbers(self,counter,words):
if counter+1<len(words):
self.translateNumbers(counter+1,words)
else:
if counter==len(words):
return True
k=0
while k<len(words[counter]):
if (not (ord(words[counter][k])>47 and ord(words[counter][k])<58)):
break
k+=1
if (k!=len(words[counter]) or k==0):
return 1
number=int(words[counter])
from copy import copy
if number==0:
self.translateNumbers(counter+1,copy(words[0:counter]+["zero"]+words[counter+1:len(words)]))
self.next.append(copy(words[0:counter]+["zero"]+words[counter+1:len(words)]))
return 1
if number<10000:
self.translateNumbers(counter+1,copy(words[0:counter]
+self.seperatewords(self.threeDigitsNumber(number))
+words[counter+1:len(words)]))
self.next.append(copy(words[0:counter]
+self.seperatewords(self.threeDigitsNumber(number))
+words[counter+1:len(words)]))
if number>999:
snumber=""
if number>1000000000:
snumber+=self.threeDigitsNumber(number/1000000000)+" billion "
number=number%1000000000
if number>1000000:
snumber+=self.threeDigitsNumber(number/1000000)+" million "
number=number%1000000
if number>1000:
snumber+=self.threeDigitsNumber(number/1000)+" thousand "
number=number%1000
snumber+=self.threeDigitsNumber(number)
self.translateNumbers(counter+1,copy(words[0:counter]+self.seperatewords(snumber)
+words[counter+1:len(words)]))
self.next.append(copy(words[0:counter]+self.seperatewords(snumber)
+words[counter+1:len(words)]))
This solution doesn't attempt to account for trailing spaces, but it is pretty fast.
typedef const char* cstring;
using std::string;
using std::endl;
std::ostream& GetOnes(std::ostream &output, int onesValue)
{
cstring ones[] = { "zero", "one", "two", "three", "four", "five", "six",
"seven", "eight", "nine" };
output << ones[onesValue];
return output;
}
std::ostream& GetSubMagnitude(std::ostream &output, int subMagnitude)
{
cstring tens[] = { "zeroty", "ten", "twenty", "thirty", "fourty", "fifty",
"sixty", "seventy", "eighty", "ninety"};
if (subMagnitude / 100 != 0)
{
GetOnes(output, subMagnitude / 100) << " hundred ";
GetSubMagnitude(output, subMagnitude - subMagnitude / 100 * 100);
}
else
{
if (subMagnitude >= 20)
{
output << tens[subMagnitude / 10] << " ";
GetOnes(output, subMagnitude - subMagnitude / 10 * 10);
}
else if (subMagnitude >= 10)
{
cstring teens[] = { "ten", "eleven", "twelve", "thirteen",
"fourteen", "fifteen", "sixteen", "seventeen",
"eighteen", "nineteen" };
output << teens[subMagnitude - 10] << " ";
}
else
{
GetOnes(output, subMagnitude) << " ";
}
}
return output;
}
std::ostream& GetLongNumber(std::ostream &output, double input)
{
cstring magnitudes[] = {"", "hundred", "thousand", "million", "billion",
"trillion"};
double magnitudeTests[] = {1, 100.0, 1000.0, 1000000.0, 1000000000.0,
1000000000000.0 };
int magTestIndex = 0;
while (floor(input / magnitudeTests[magTestIndex++]) != 0);
magTestIndex -= 2;
if (magTestIndex >= 0)
{
double subMagnitude = input / magnitudeTests[magTestIndex];
GetSubMagnitude(output, (int)subMagnitude);
if (magTestIndex) {
output << magnitudes[magTestIndex] << " ";
double remainder = input - (floor(input /
magnitudeTests[magTestIndex]) *
magnitudeTests[magTestIndex]);
if (floor(remainder) > 0)
{
GetLongNumber(output, remainder);
}
}
}
else
{
output << "zero";
}
return output;
}
Start by solving 1-99, using a list of numbers for 1-20, and then 30, 40, ..., 90. Then add hundreds to get 1-999. Then use that routine to give the number of each power of 1,000 for as high as you want to go (I think the highest standard nomenclature is for decillion, which is 10^33).
One slight caveat is that it's a little tricky to get the blanks right in all cases if you're trying to start and end without an excess blank. The easy solution is to put a blank after every word, and then strip off the trailing blank when you're all done. If you try to be more precise while building the string, you're likely to end up with missing blanks or excess blanks.
.) make a library of all numbers & positions (e.g. 1 has other notation than 10, another than 100 etc)
.) make a list of exceptions (e.g. for 12) and be aware, that in your algorythm, the same exception are for 112, 1012 etc.
if you want even more speed, make a cached set of numbers that you need.
Note some rules:
Tens numbers (twenty, thirty, etc.) ending in y are followed by hyphens.
Teens are special (except 15-19, but they're still special).
Everything else is just some combination of digit place like "three thousand".
You can get the place of a number by using floor division of integers: 532 / 100 -> 5
This is part of Common Lisp!
Here's how GNU CLISP does it, and here's how CMUCL does it (easier to read, IMHO).
Doing a code search for "format million billion" will turn up lots of them.
Here's a coding problem for those that like this kind of thing. Let's see your implementations (in your language of choice, of course) of a function which returns a human readable String representation of a specified Integer. For example:
humanReadable(1) returns "one".
humanReadable(53) returns "fifty-three".
humanReadable(723603) returns "seven hundred and twenty-three thousand, six hundred and three".
humanReadable(1456376562) returns "one billion, four hundred and fifty-six million, three hundred and seventy-six thousand, five hundred and sixty-two".
Bonus points for particularly clever/elegant solutions!
It might seem like a pointless exercise, but there are number of real world applications for this kind of algorithm (although supporting numbers as high as a billion may be overkill :-)
There was already a question about this:
Convert integers to written numbers
The answer is for C#, but I think you can figure it out.
import math
def encodeOnesDigit(num):
return ['', 'one', 'two', 'three', 'four', 'five', 'six', 'seven', 'eight', 'nine'][num]
def encodeTensDigit(num):
return ['twenty', 'thirty', 'forty', 'fifty', 'sixty', 'seventy', 'eighty', 'ninety'][num-2]
def encodeTeens(num):
if num < 10:
return encodeOnesDigit(num)
else:
return ['ten', 'eleven', 'twelve', 'thirteen', 'fourteen', 'fifteen', 'sixteen', 'seventeen', 'eighteen', 'nineteen'][num-10]
def encodeTriplet(num):
if num == 0: return ''
str = ''
if num >= 100:
str = encodeOnesDigit(num / 100) + ' hundred'
tens = num % 100
if tens >= 20:
if str != '': str += ' '
str += encodeTensDigit(tens / 10)
if tens % 10 > 0:
str += '-' + encodeOnesDigit(tens % 10)
elif tens != 0:
if str != '': str += ' '
str += encodeTeens(tens)
return str
def zipNumbers(numList):
if len(numList) == 1:
return numList[0]
strList = ['', ' thousand', ' million', ' billion'] # Add more as needed
strList = strList[:len(numList)]
strList.reverse()
joinedList = zip(numList, strList)
joinedList = [item for item in joinedList if item[0] != '']
return ', '.join(''.join(item) for item in joinedList)
def humanReadable(num):
if num == 0: return 'zero'
negative = False
if num < 0:
num *= -1
negative = True
numString = str(num)
tripletCount = int(math.ceil(len(numString) / 3.0))
numString = numString.zfill(tripletCount * 3)
tripletList = [int(numString[i*3:i*3+3]) for i in range(tripletCount)]
readableList = [encodeTriplet(num) for num in tripletList]
readableStr = zipNumbers(readableList)
return 'negative ' + readableStr if negative else readableStr
Supports up to 999 million, but no negative numbers:
String humanReadable(int inputNumber) {
if (inputNumber == -1) {
return "";
}
int remainder;
int quotient;
quotient = inputNumber / 1000000;
remainder = inputNumber % 1000000;
if (quotient > 0) {
return humanReadable(quotient) + " million, " + humanReadable(remainder);
}
quotient = inputNumber / 1000;
remainder = inputNumber % 1000;
if (quotient > 0) {
return humanReadable(quotient) + " thousand, " + humanReadable(remainder);
}
quotient = inputNumber / 100;
remainder = inputNumber % 100;
if (quotient > 0) {
return humanReadable(quotient) + " hundred, " + humanReadable(remainder);
}
quotient = inputNumber / 10;
remainder = inputNumber % 10;
if (remainder == 0) {
//hackish way to flag the algorithm to not output something like "twenty zero"
remainder = -1;
}
if (quotient == 1) {
switch(inputNumber) {
case 10:
return "ten";
case 11:
return "eleven";
case 12:
return "twelve";
case 13:
return "thirteen";
case 14:
return "fourteen";
case 15:
return "fifteen";
case 16:
return "sixteen";
case 17:
return "seventeen";
case 18:
return "eighteen";
case 19:
return "nineteen";
}
}
switch(quotient) {
case 2:
return "twenty " + humanReadable(remainder);
case 3:
return "thirty " + humanReadable(remainder);
case 4:
return "forty " + humanReadable(remainder);
case 5:
return "fifty " + humanReadable(remainder);
case 6:
return "sixty " + humanReadable(remainder);
case 7:
return "seventy " + humanReadable(remainder);
case 8:
return "eighty " + humanReadable(remainder);
case 9:
return "ninety " + humanReadable(remainder);
}
switch(inputNumber) {
case 0:
return "zero";
case 1:
return "one";
case 2:
return "two";
case 3:
return "three";
case 4:
return "four";
case 5:
return "five";
case 6:
return "six";
case 7:
return "seven";
case 8:
return "eight";
case 9:
return "nine";
}
}
using System;
namespace HumanReadable
{
public static class HumanReadableExt
{
private static readonly string[] _digits = {
"", "one", "two", "three", "four", "five",
"six", "seven", "eight", "nine", "eleven", "twelve",
"thirteen", "fourteen", "fifteen", "sixteen", "seventeen",
"eighteen", "nineteen"
};
private static readonly string[] _teens = {
"", "", "twenty", "thirty", "forty", "fifty",
"sixty", "seventy", "eighty", "ninety"
};
private static readonly string[] _illions = {
"", "thousand", "million", "billion", "trillion"
};
private static string Seg(int number)
{
var work = string.Empty;
if (number >= 100)
work += _digits[number / 100] + " hundred ";
if ((number % 100) < 20)
work += _digits[number % 100];
else
work += _teens[(number % 100) / 10] + "-" + _digits[number % 10];
return work;
}
public static string HumanReadable(this int number)
{
if (number == 0)
return "zero";
var work = string.Empty;
var parts = new string[_illions.Length];
for (var ind = 0; ind < parts.Length; ind++)
parts[ind] = Seg((int) (number % Math.Pow(1000, ind + 1) / Math.Pow(1000, ind)));
for (var ind = 0; ind < parts.Length; ind++)
if (!string.IsNullOrEmpty(parts[ind]))
work = parts[ind] + " " + _illions[ind] + ", " + work;
work = work.TrimEnd(',', ' ');
var lastSpace = work.LastIndexOf(' ');
if (lastSpace >= 0)
work = work.Substring(0, lastSpace) + " and" + work.Substring(lastSpace);
return work;
}
}
class Program
{
static void Main(string[] args)
{
Console.WriteLine(1.HumanReadable());
Console.WriteLine(53.HumanReadable());
Console.WriteLine(723603.HumanReadable());
Console.WriteLine(1456376562.HumanReadable());
Console.ReadLine();
}
}
}
There's one huge problem about this function implementation. It is it's future localization. That function, written by english native speaker, most probably wouldn't work right for any other language than english. It is nearly impossible to write general easy localizable function for any human language dialect in a world, unless you really need to keep it general. Actually in real world you do not need to operate with huge integer numbers, so you can just keep all the numbers in a big (or even not so big) string array.
agreed that there are a number of real world applications.
as such there's already a number of real world implementations.
it's been part of bsdgames since pretty much forever...
> man number