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I'm having trouble figuring out the type of problem this is. I'm still a student and haven't taken a graph theory/linear optimization class yet.
The only thing I know for sure is to check for negative cycles, as this means you can rack the resource limit up to infinity, allowing for you to pick up each rabbit. I don't know the "reason" to pick the next path. I also don't know when to terminate, as you could keep using all of the edges and make the resource limit drop below 0 forever, but never escape.
I'm not really looking for code (as this is a coding challenge), only the type of problem this is (Ex: Max Flow, Longest Path, Shortest Path, etc.) If you an algorithm that fits this already that would be extra awesome. Thanks.
The time it takes to move from your starting point to all of the bunnies and to the bulkhead will be given to you in a square matrix of integers. Each row will tell you the time it takes to get to the start, first bunny, second bunny, ..., last bunny, and the bulkhead in that order. The order of the rows follows the same pattern (start, each bunny, bulkhead). The bunnies can jump into your arms, so picking them up is instantaneous, and arriving at the bulkhead at the same time as it seals still allows for a successful, if dramatic, escape. (Don't worry, any bunnies you don't pick up will be able to escape with you since they no longer have to carry the ones you did pick up.) You can revisit different spots if you wish, and moving to the bulkhead doesn't mean you have to immediately leave - you can move to and from the bulkhead to pick up additional bunnies if time permits.
In addition to spending time traveling between bunnies, some paths interact with the space station's security checkpoints and add time back to the clock. Adding time to the clock will delay the closing of the bulkhead doors, and if the time goes back up to 0 or a positive number after the doors have already closed, it triggers the bulkhead to reopen. Therefore, it might be possible to walk in a circle and keep gaining time: that is, each time a path is traversed, the same amount of time is used or added.
Write a function of the form answer(times, time_limit) to calculate the most bunnies you can pick up and which bunnies they are, while still escaping through the bulkhead before the doors close for good. If there are multiple sets of bunnies of the same size, return the set of bunnies with the lowest prisoner IDs (as indexes) in sorted order. The bunnies are represented as a sorted list by prisoner ID, with the first bunny being 0. There are at most 5 bunnies, and time_limit is a non-negative integer that is at most 999.
It's a planning problem, basically. The generic approach to planning is to identify the possible states of the world, the initial state, transitions between states, and the final states. Then search the graph that this data imply, most simply using breadth-first search.
For this problem, the relevant state is (1) how much time is left (2) which rabbits we've picked up (3) where we are right now. This means 1,000 clock settings (I'll talk about added time in a minute) times 2^5 = 32 subsets of bunnies times 7 positions = 224,000 possible states, which is a lot for a human but not a computer.
We can deal with added time by swiping a trick from Johnson's algorithm. As Tymur suggests in a comment, run Bellman--Ford and either find a negative cycle (in which case all rabbits can be saved by running around the negative cycle enough times first) or potentials that, when applied, make all times nonnegative. Don't forget to adjust the starting time by the difference in potential between the starting position and the bulkhead.
There you go. I started Google Foobar yesterday. I'll be starting Level 5 shortly. This was my 2nd problem here at level 4. The solution is fast enough as I tried memoizing the states without using the utils class. Anyway, loved the experience. This was by far the best problem solved by me since I got to use Floyd-Warshall(to find the negative cycle if it exists), Bellman-Ford(as a utility function to the weight readjustment step used popularly in algorithms like Johnson's and Suurballe's), Johnson(weight readjustment!), DFS(for recursing over steps) and even memoization using a self-designed hashing function :)
Happy Coding!!
public class Solution
{
public static final int INF = 100000000;
public static final int MEMO_SIZE = 10000;
public static int[] lookup;
public static int[] lookup_for_bunnies;
public static int getHashValue(int[] state, int loc)
{
int hashval = 0;
for(int i = 0; i < state.length; i++)
hashval += state[i] * (1 << i);
hashval += (1 << loc) * 100;
return hashval % MEMO_SIZE;
}
public static boolean findNegativeCycle(int[][] times)
{
int i, j, k;
int checkSum = 0;
int V = times.length;
int[][] graph = new int[V][V];
for(i = 0; i < V; i++)
for(j = 0; j < V; j++)
{
graph[i][j] = times[i][j];
checkSum += times[i][j];
}
if(checkSum == 0)
return true;
for(k = 0; k < V; k++)
for(i = 0; i < V; i++)
for(j = 0; j < V; j++)
if(graph[i][j] > graph[i][k] + graph[k][j])
graph[i][j] = graph[i][k] + graph[k][j];
for(i = 0; i < V; i++)
if(graph[i][i] < 0)
return true;
return false;
}
public static void dfs(int[][] times, int[] state, int loc, int tm, int[] res)
{
int V = times.length;
if(loc == V - 1)
{
int rescued = countArr(state);
int maxRescued = countArr(res);
if(maxRescued < rescued)
for(int i = 0; i < V; i++)
res[i] = state[i];
if(rescued == V - 2)
return;
}
else if(loc > 0)
state[loc] = 1;
int hashval = getHashValue(state, loc);
if(tm < lookup[hashval])
return;
else if(tm == lookup[hashval] && countArr(state) <= lookup_for_bunnies[loc])
return;
else
{
lookup_for_bunnies[loc] = countArr(state);
lookup[hashval] = tm;
for(int i = 0; i < V; i++)
{
if(i != loc && (tm - times[loc][i]) >= 0)
{
boolean stateCache = state[i] == 1;
dfs(times, state, i, tm - times[loc][i], res);
if(stateCache)
state[i] = 1;
else
state[i] = 0;
}
}
}
}
public static int countArr(int[] arr)
{
int counter = 0;
for(int i = 0; i < arr.length; i++)
if(arr[i] == 1)
counter++;
return counter;
}
public static int bellmanFord(int[][] adj, int times_limit)
{
int V = adj.length;
int i, j, k;
int[][] graph = new int[V + 1][V + 1];
for(i = 1; i <= V; i++)
graph[i][0] = INF;
for(i = 0; i < V; i++)
for(j = 0; j < V; j++)
graph[i + 1][j + 1] = adj[i][j];
int[] distance = new int[V + 1] ;
for(i = 1; i <= V; i++)
distance[i] = INF;
for(i = 1; i <= V; i++)
for(j = 0; j <= V; j++)
{
int minDist = INF;
for(k = 0; k <= V; k++)
if(graph[k][j] != INF)
minDist = Math.min(minDist, distance[k] + graph[k][j]);
distance[j] = Math.min(distance[j], minDist);
}
for(i = 0; i < V; i++)
for(j = 0; j < V; j++)
adj[i][j] += distance[i + 1] - distance[j + 1];
return times_limit + distance[1] - distance[V];
}
public static int[] solution(int[][] times, int times_limit)
{
int V = times.length;
if(V == 2)
return new int[]{};
if(findNegativeCycle(times))
{
int ans[] = new int[times.length - 2];
for(int i = 0; i < ans.length; i++)
ans[i] = i;
return ans;
}
lookup = new int[MEMO_SIZE];
lookup_for_bunnies = new int[V];
for(int i = 0; i < V; i++)
lookup_for_bunnies[i] = -1;
times_limit = bellmanFord(times, times_limit);
int initial[] = new int[V];
int res[] = new int[V];
dfs(times, initial, 0, times_limit, res);
int len = countArr(res);
int ans[] = new int[len];
int counter = 0;
for(int i = 0; i < res.length; i++)
if(res[i] == 1)
{
ans[counter++] = i - 1;
if(counter == len)
break;
}
return ans;
}
}
Find minimum cost of tickets required to buy for traveling on known days of the month (1...30). Three types of tickets are available : 1-day ticket valid for 1 days and costs 2 units, 7-days ticket valid for 7 days and costs 7 units, 30-days ticket valid for 30 days and costs 25 units.
For eg: I want to travel on [1,4,6,7,28,30] days of the month i.e. 1st, 4th, 6th ... day of the month. How to buy tickets so that the cost is minimum.
I tried to use dynamic programming to solve this but the solution is not giving me the correct answer for all cases. Here is my solution in Java :
public class TicketsCost {
public static void main(String args[]){
int[] arr = {1,5,6,9,28,30};
System.out.println(findMinCost(arr));
}
public static int findMinCost(int[] arr) {
int[][] dp = new int[arr.length][3];
int[] tDays = {1,7,30};
int[] tCost = {2,7,25};
for (int i = 0; i < arr.length; i++) {
for (int j = 0; j < 3; j++) {
if (j==0){
dp[i][j]= (i+1)*tCost[j];
}
else{
int c = arr[i]-tDays[j];
int tempCost = tCost[j];
int k;
if (c>=arr[0] && i>0){
for (k = i-1; k >= 0; k--) {
if (arr[k]<=c){
c = arr[k];
}
}
tempCost += dp[c][j];
int tempCostX = dp[i-1][j] + tCost[0];
tempCost = Math.min(tempCost,tempCostX);
}
dp[i][j] = Math.min(tempCost,dp[i][j-1]);
}
}
}
return dp[arr.length-1][2];
}
}
The solution doesn't work for {1,7,8,9,10} input, it gives 10 but the correct answer should be 9. Also, for {1,7,8,9,10,15} it give 13 but the correct is 11.
I have posted my solution not for other to debug it for me but just for reference. I was taken a bottom-up dynamic programming approach for this problem. Is this approach correct?
Let MC(d) denote the minimum cost that will pay for all trips on days 1 through d. The desired answer is then MC(30).
To calculate MC(d), observe the following:
If there's no trip on day d, then MC(d) = MC(d − 1).
As a special case, MC(d) = 0 for all d ≤ 0.
Otherwise, the minimum cost involves one of the following:
A 1-day pass on day d. In this case, MC(d) = MC(d − 1) + 2.
A 7-day pass ending on or after day d. In this case, MC(d) = min(MC(d − 7), MC(d − 6), …, MC(d − 1)) + 7.
And since MC is nondecreasing (adding a day never reduces the minimum cost), this can be simplified to MC(d) = MC(d − 7) + 7. (Hat-tip to Ravi for pointing this out.)
A 30-day pass covering the whole period. In this case, MC(d) = 25.
As you've realized, dynamic programming (bottom-up recursion) is well-suited to this.
For ease of coding, I suggest we start by converting the list of days into a lookup table for "is this a trip day?":
boolean[] isDayWithTrip = new boolean[31]; // note: initializes to false
for (final int dayWithTrip : arr) {
isDayWithTrip[dayWithTrip] = true;
}
We can then create an array to track the minimum costs, and populate it starting from index 0:
int[] minCostUpThroughDay = new int[31];
minCostUpThroughDay[0] = 0; // technically redundant
for (int d = 1; d <= 30; ++d) {
if (! isDayWithTrip[d]) {
minCostUpThroughDay[d] = minCostUpThroughDay[d-1];
continue;
}
int minCost;
// Possibility #1: one-day pass on day d:
minCost = minCostUpThroughDay[d-1] + 2;
// Possibility #2: seven-day pass ending on or after day d:
minCost =
Math.min(minCost, minCostUpThroughDay[Math.max(0, d-7)] + 7);
// Possibility #3: 30-day pass for the whole period:
minCost = Math.min(minCost, 25);
minCostUpThroughDay[d] = minCost;
}
And minCostUpThroughDay[30] is the result.
You can see the above code in action at: https://ideone.com/1Xx1fd.
One recursive solution in Python3.
from typing import List
def solution(A: List[int]) -> int:
if not any(A):
return 0
tickets = {
1: 2,
7: 7,
30: 25,
}
import sys
min_cost = sys.maxsize
size = len(A)
for length, price in tickets.items():
current_cost = price
idx = 0
last_day = A[idx] + length
while idx < size and A[idx] < last_day:
idx += 1
if current_cost > min_cost:
continue
current_cost += solution(A[idx:])
if current_cost < min_cost:
min_cost = current_cost
return min_cost
if __name__ == '__main__':
cases = {
11: [1, 4, 6, 7, 28, 30],
9: [1, 7, 8, 9, 10],
}
for expect, parameters in cases.items():
status = (expect == solution(parameters))
print("case pass status: %s, detail: %s == solution(%s)" %
(status, expect, parameters))
public class Main03v3
{
public static void main(String[] args)
{
int[] A = {1,7,8,9,10,15,16,17,18,21,25};
System.out.println("Traveling days:\r\n "+Arrays.toString(A));
int cost = solution(A);
System.out.println("\r\nMinimum cost is " + cost);
System.out.println("\r\n" + new String(new char[40]).replace("\0", "-"));
}
public static int solution(int[] A)
{
if (A == null) return -1;
int sevenDays = 7;
int dayCost = 2, weekCost = 7, monthCost = 25;
int ratio_WeekAndDays = weekCost / dayCost;
int len = A.length;
if (len == 0) return -1;
if (len <= 3) return len * dayCost;
int cost[] = new int[len];
int i = 0;
while (i < len)
{
int startIdx = i, endIdx = i + 1;
while (endIdx < len && A[endIdx]-A[startIdx] < sevenDays)
endIdx++;
if (endIdx-startIdx > ratio_WeekAndDays)
{
if (endIdx >= startIdx + sevenDays)
endIdx = startIdx + sevenDays;
int j = startIdx;
cost[j] = ((j == 0) ? 0 : cost[j-1]) + weekCost;
while (++j < endIdx) {
cost[j] = cost[j-1];
}
i = j;
}
else
{
cost[i] = ((i == 0) ? 0 : cost[i-1]) + dayCost;
i++;
}
}
int finalCost = Math.min(cost[len-1], monthCost);
return finalCost;
}
}
Find minimum cost of tickets in JavaScript
case 1 : if input is [1,7,8,9,10] then the required output is 9
case 2 : if input is [1,7,8,9,10,15] then the required output is 11
function calMinCosts(arr){
if(!arr || arr.length===0)
return 0;
var len = arr.length;
var costsOfDateArr = Array.apply(null,{length:arr[len-1]+1}).map(()=>0);
var price1=2,price2=7,price3=25;
var days=7;
var index=0,n=costsOfDateArr.length;
for(var i=1;i<n;i++){
if(i===arr[index]){
if(i>=days+1){
costsOfDateArr[i] = Math.min(costsOfDateArr[i-days-1]+price2, costsOfDateArr[i-1]+price1);
}else{
costsOfDateArr[i] = Math.min(costsOfDateArr[0]+price2, costsOfDateArr[i-1]+price1);
}
index+=1;
}else{
costsOfDateArr[i] = costsOfDateArr[i-1];
}
}
return Math.min(price3,costsOfDateArr[n-1]);
}
console.log(calMinCosts([1,7,8,9,10]))
console.log(calMinCosts([1,7,8,9,10,15]))
Here is the C++ solution including print outs
#include <vector>
#include <iostream>
#include <cmath>
#include <algorithm>
int compute(std::vector<int> &A)
{
int sum[A.size()][A.size()+1];
for (int i = 0; i < A.size(); i++)
{
for(int j =0; j < A.size(); j++)
{
sum[i][j]=2;
}
}
for (int k = 0; k < A.size();k++)
{
sum[k][A.size()]=0;
}
for (int i = 0; i < A.size(); i++)
{
for(int j = 0; j < A.size(); j++)
{
if (i!=j)
{
if (sum[i][i] != 7)
{
int temp = abs(A[j]-A[i]);
if (temp<7 && abs(j-i)>=3)
{
sum[i][i]=7;
sum[i][j]=7;
if (i>j)
{
for(int k = j;k < i;k++)
sum[i][k]=7;
}
else
{
for(int k = i;k < j;k++)
sum[i][k]=7;
}
}
}
}
}
}
for (int i = 0; i < A.size(); ++i)
{
for(int j = 0; j < A.size(); ++j)
{
if (sum[i][j]==7)
{
sum[i][A.size()]+=1;
}
}
}
for (int i = 0; i < A.size(); ++i)
{
for (int j = 0; j < A.size()+1; ++j)
std::cout<<sum[i][j]<<" ";
std::cout<<std::endl;
}
int result = 0;
int row = A.size()-1;
int column = A.size()-1;
while(1)
{
int value = sum[row][A.size()];
if (value == 0)
value=1;
int temp = sum[row][column];
result += temp;
row = row-value;
column = column-value;
while (sum[row][column+1]==7 && row>=0)
{
row-=1;
column-=1;
result+=2;
}
if (row < 0)
break;
}
return result;
}
int solution(std::vector<int> &A) {
if (A.size() > 24)
return 25;
if (A.size() <= 3)
return A.size() * 2;
return std::min(25,compute(A));
}
int main()
{
std::vector<int> AA={1,2,3,4,5,29,30};
std::vector<int> B={1,2,3,4,5};
std::vector<int> A={1,2,3,4,5,9,10,11,12,13,14,17,18,20,21};
std::vector<int> C={1,2,3,12};
std::vector<int> D={1,2,3,4,12,13,14,15,29,30};
std::vector<int> DD={1,2,3,4,5,14,17,18,19,20,23,28,29,30};
std::vector<int> CC={1,2,3,4,5,6,7,9,14,17,18,19,20,23,28,29,30};
std::cout<<solution(AA)<<std::endl;
std::cout<<solution(D)<<std::endl;
std::cout<<solution(B)<<std::endl;
std::cout<<solution(A)<<std::endl;
std::cout<<solution(C)<<std::endl;
std::cout<<solution(DD)<<std::endl;
std::cout<<solution(CC)<<std::endl;
return 0;
}
Solved using the same approach of bottom-up dynamic programming. Here is the full solution :
public class PublicTicketCost {
public static void main(String args[]){
int[] arr = {1,7,8,9,10,15,16,17,18,21,25};
int[] tDays = {1,7,30};
int[] tCost = {2,7,25};
System.out.println(minCost(arr, tDays, tCost));
}
public static int minCost(int[] arr, int[] tDays, int[] tCost) {
int[][] dp = new int[arr.length][tDays.length];
for (int i = 0; i < arr.length; i++) {
for (int j = 0; j < tDays.length; j++) {
int prevDayIndex = findPrevDayIndex(arr,i,tDays,j);
int prevCost = prevDayIndex>=0 ? dp[prevDayIndex][tDays.length-1] : 0;
int currCost = prevCost + tCost[j];
if(j-1>=0){
currCost = Math.min(currCost, dp[i][j-1]);
}
dp[i][j] = currCost;
}
}
//print(dp);
return dp[arr.length-1][tDays.length-1];
}
private static void print(int arr[][]){
for (int i = 0; i < arr.length; i++) {
for (int j = 0; j < arr[0].length; j++) {
System.out.print(arr[i][j]+" ");
}
System.out.println();
}
}
private static int findPrevDayIndex(int[] arr, int i, int[] days, int j){
int validAfterDate = arr[i] - days[j];
if (validAfterDate<1){
return -1;
}
for (int k = i-1; k >= 0; k--) {
if (arr[k]<=validAfterDate){
return k;
}
}
return -1;
}
}
http://ideone.com/sfgxGo
After looking at the common stair climbing problem, I started to wonder if this could be abstracted to a function that allows for the entry of both number of stairs and the max increment steps allowed as parameters.
I would like to be able to write a function with this signature. If max_step_increment is 4, that means the stair-climber could takes steps of 1, 2, 3, or 4 at a time.
def stair_paths(num_steps, max_step_increment):
...
return answer
I would call this function like stair_paths(10, 4).
Solved in Java. This would be more elegant if your method declaration were:
int stairPaths(int numSteps, int maxStepIncrement)
As you have defined it, here is the dynamic programming solution:
int stairPaths(int numSteps, int... stepsAllowed)
{
if (stepsAllowed.length == 0) {
return 0;
}
Arrays.sort(stepsAllowed);
if (stepsAllowed[0] < 1) {
throw new IllegalArgumentException("Invalid step increment " + stepsAllowed[0]);
}
int maxStepIncrement = stepsAllowed[stepsAllowed.length - 1];
int[] priorElements = new int[maxStepIncrement];
priorElements[maxStepIncrement - 1] = 1;
priorElements[maxStepIncrement - 2] = 1;
for (int i = 2; i <= numSteps; i++) {
int nextElement = 0;
for (int j = 0; j < stepsAllowed.length; j++) {
nextElement += priorElements[maxStepIncrement - stepsAllowed[j]];
}
for (int k = 1; k < maxStepIncrement; k++) {
priorElements[k - 1] = priorElements[k];
}
priorElements[maxStepIncrement - 1] = nextElement;
}
return priorElements[maxStepIncrement - 1];
}
Let f[n] means the number of ways to get to n stairs with all allowed steps.
Initially, f[0]=1, the remaining are all 0.
Then, f[i]=sigma(f[i-allowedSteps[j]]), where allowedSteps[j] are all possible allowed steps.
And the final answer should be f[numStairs], which is f[10] in your example.
I ran into an implementation problem when trying to solve this classic problem using DP.
The problem is given a set of coins, and return the number of ways of making a change.
The DP equation is something like the following:
DP[i] += DP[i - coin[j]]
where DP[i] means the number of ways of making change for i.
Here is a straightforward implementation, which is incorrect:
int make_change_wrong(int coin[], int size, int change) {
vector<int> DP(change + 1, 0);
DP[0] = 1;
for (int i = 1; i <= change; ++i) {
for (int j = 0; j < size; ++j) {
if (i - coin[j] >= 0 ) {
DP[i] += DP[i - coin[j]];
}
}
}
return DP[change];
}
Given input:
int coin[] = {1, 5}
change = 6.
make_change_wrong(coin, 2, 6) returns 3, but 2 is correct.
Using the same logic, I re-write it in a less intuitive way and get the correct answer:
int make_change(int coin[], int size, int change) {
vector<int> DP(change + 1, 0);
DP[0] = 1;
for (int i = 0; i < size; ++i) {
for (int j = coin[i]; j <= change; ++j) {
DP[j] += DP[j - coin[i]];
}
}
return DP[change];
}
This puzzled me a lot because to me, they're the same thing...
Can someone illustrate a bit on the problems in the two implementations?
Your first algorithm is wrong.
DP[5] = 2 {1,1,1,1,1}, {5}
DP[6] = DP[5] + DP[1] = 3
you are counting {5,1} twice.
EDITED
So the standard trick for doing this is that you keep a count of the denomination you are allowed to use
DP[i,m] = DP[i-coin[m],m] + DP[i,m-1]
which means number of ways of making a change of i amount using coins in range[1..m].
This is obviously, you either use the mth denomination or you don't.
The second algorithm you are using is doing the same trick but is a really clever way to do that, take the ith coin and see what all change you can generate using it. This will avoid over counting because you avoid doing things like {1,5} and {5,1}.
This problem is in the interview prep book Cracking the Coding Interview, and the solution given in the book is not optimized at all. It uses recursion (without DP) to calculate sub-problems repeatedly and therefore runs in O(N^3) which is especially ironic since it's part of the Dynamic Programming chapter.
Here's a very simple working solution (Java) which uses DP and runs in O(N) time.
static int numCombos(int n) {
int[] dyn = new int[n + 1];
Arrays.fill(dyn, 0);
dyn[0] = 1;
for (int i = 1; i <= n; i++) dyn[i] += dyn[i - 1];
for (int i = 5; i <= n; i++) dyn[i] += dyn[i - 5];
for (int i = 10; i <= n; i++) dyn[i] += dyn[i - 10];
for (int i = 25; i <= n; i++) dyn[i] += dyn[i - 25];
return dyn[n];
}
Please try your input for your second method:
coin[5] = {1,5,10,20,30};
make_change(coin,5,30);
It returns 21. Please check my test case.
Input: A 2-dimensional array NxN - Matrix - with positive and negative elements.Output: A submatrix of any size such that its summation is the maximum among all possible submatrices.
Requirement: Algorithm complexity to be of O(N^3)
History: With the help of the Algorithmist, Larry and a modification of Kadane's Algorithm, i managed to solve the problem partly which is determining the summation only - below in Java.
Thanks to Ernesto who managed to solve the rest of the problem which is determining the boundaries of the matrix i.e. top-left, bottom-right corners - below in Ruby.
Here's an explanation to go with the posted code. There are two key tricks to make this work efficiently: (I) Kadane's algorithm and (II) using prefix sums. You also need to (III) apply the tricks to the matrix.
Part I: Kadane's algorithm
Kadane's algorithm is a way to find a contiguous subsequence with maximum sum. Let's start with a brute force approach for finding the max contiguous subsequence and then consider optimizing it to get Kadane's algorithm.
Suppose you have the sequence:
-1, 2, 3, -2
For the brute force approach, walk along the sequence generating all possible subsequences as shown below. Considering all possibilities, we can start, extend, or end a list with each step.
At index 0, we consider appending the -1
-1, 2, 3, -2
^
Possible subsequences:
-1 [sum -1]
At index 1, we consider appending the 2
-1, 2, 3, -2
^
Possible subsequences:
-1 (end) [sum -1]
-1, 2 [sum 1]
2 [sum 2]
At index 2, we consider appending the 3
-1, 2, 3, -2
^
Possible subsequences:
-1, (end) [sum -1]
-1, 2 (end) [sum -1]
2 (end) [sum 2]
-1, 2, 3 [sum 4]
2, 3 [sum 5]
3 [sum 3]
At index 3, we consider appending the -2
-1, 2, 3, -2
^
Possible subsequences:
-1, (end) [sum -1]
-1, 2 (end) [sum 1]
2 (end) [sum 2]
-1, 2 3 (end) [sum 4]
2, 3 (end) [sum 5]
3, (end) [sum 3]
-1, 2, 3, -2 [sum 2]
2, 3, -2 [sum 3]
3, -2 [sum 1]
-2 [sum -2]
For this brute force approach, we finally pick the list with the best sum, (2, 3), and that's the answer. However, to make this efficient, consider that you really don't need to keep every one of the lists. Out of the lists that have not ended, you only need to keep the best one, the others cannot do any better. Out of the lists that have ended, you only might need to keep the best one, and only if it's better than ones that have not ended.
So, you can keep track of what you need with just a position array and a sum array. The position array is defined like this: position[r] = s keeps track of the list which ends at r and starts at s. And, sum[r] gives a sum for the subsequence ending at index r. This is optimized approach is Kadane's algorithm.
Running through the example again keeping track of our progress this way:
At index 0, we consider appending the -1
-1, 2, 3, -2
^
We start a new subsequence for the first element.
position[0] = 0
sum[0] = -1
At index 1, we consider appending the 2
-1, 2, 3, -2
^
We choose to start a new subsequence because that gives a higher sum than extending.
position[0] = 0 sum[0] = -1
position[1] = 1 sum[1] = 2
At index 2, we consider appending the 3
-1, 2, 3, -2
^
We choose to extend a subsequence because that gives a higher sum than starting a new one.
position[0] = 0 sum[0] = -1
position[1] = 1 sum[1] = 2
position[2] = 1 sum[2] = 5
Again, we choose to extend because that gives a higher sum that starting a new one.
-1, 2, 3, -2
^
position[0] = 0 sum[0] = -1
position[1] = 1 sum[1] = 2
position[2] = 1 sum[2] = 5
positions[3] = 3 sum[3] = 3
Again, the best sum is 5 and the list is from index 1 to index 2, which is (2, 3).
Part II: Prefix sums
We want to have a way to compute the sum along a row, for any start point to any endpoint. I want to compute that sum in O(1) time rather than just adding, which takes O(m) time where m is the number of elements in the sum. With some precomputing, this can be achieved. Here's how. Suppose you have a matrix:
a d g
b e h
c f i
You can precompute this matrix:
a d g
a+b d+e g+h
a+b+c d+e+f g+h+i
Once that is done you can get the sum running along any column from any start to endpoint in the column just by subtracting two values.
Part III: Bringing tricks together to find the max submatrix
Assume that you know the top and bottom row of the max submatrix. You could do this:
Ignore rows above your top row and ignore rows below your bottom
row.
With what matrix remains, consider the using sum of each column to
form a sequence (sort of like a row that represents multiple rows).
(You can compute any element of this sequence rapidly with the prefix
sums approach.)
Use Kadane's approach to figure out best subsequence in this
sequence. The indexes you get will tell you the left and right
positions of the best submatrix.
Now, what about actually figuring out the top and bottom row? Just try all possibilities. Try putting the top anywhere you can and putting the bottom anywhere you can, and run the Kadane-base procedure described previously for every possibility. When you find a max, you keep track of the top and bottom position.
Finding the row and column takes O(M^2) where M is the number of rows. Finding the column takes O(N) time where N is the number of columns. So total time is O(M^2 * N). And, if M=N, the time required is O(N^3).
About recovering the actual submatrix, and not just the maximum sum, here's what I got. Sorry I do not have time to translate my code to your java version, so I'm posting my Ruby code with some comments in the key parts
def max_contiguous_submatrix_n3(m)
rows = m.count
cols = rows ? m.first.count : 0
vps = Array.new(rows)
for i in 0..rows
vps[i] = Array.new(cols, 0)
end
for j in 0...cols
vps[0][j] = m[0][j]
for i in 1...rows
vps[i][j] = vps[i-1][j] + m[i][j]
end
end
max = [m[0][0],0,0,0,0] # this is the result, stores [max,top,left,bottom,right]
# these arrays are used over Kadane
sum = Array.new(cols) # obvious sum array used in Kadane
pos = Array.new(cols) # keeps track of the beginning position for the max subseq ending in j
for i in 0...rows
for k in i...rows
# Kadane over all columns with the i..k rows
sum.fill(0) # clean both the sum and pos arrays for the upcoming Kadane
pos.fill(0)
local_max = 0 # we keep track of the position of the max value over each Kadane's execution
# notice that we do not keep track of the max value, but only its position
sum[0] = vps[k][0] - (i==0 ? 0 : vps[i-1][0])
for j in 1...cols
value = vps[k][j] - (i==0 ? 0 : vps[i-1][j])
if sum[j-1] > 0
sum[j] = sum[j-1] + value
pos[j] = pos[j-1]
else
sum[j] = value
pos[j] = j
end
if sum[j] > sum[local_max]
local_max = j
end
end
# Kadane ends here
# Here's the key thing
# If the max value obtained over the past Kadane's execution is larger than
# the current maximum, then update the max array with sum and bounds
if sum[local_max] > max[0]
# sum[local_max] is the new max value
# the corresponding submatrix goes from rows i..k.
# and from columns pos[local_max]..local_max
# the array below contains [max_sum,top,left,bottom,right]
max = [sum[local_max], i, pos[local_max], k, local_max]
end
end
end
return max # return the array with [max_sum,top,left,bottom,right]
end
Some notes for clarification:
I use an array to store all the values pertaining to the result for convenience. You can just use five standalone variables: max, top, left, bottom, right. It's just easier to assign in one line to the array and then the subroutine returns the array with all the needed information.
If you copy and paste this code in a text-highlight-enabled editor with Ruby support you'll obviously understand it better. Hope this helps!
There are already plenty of answers, but here is another Java implementation I wrote. It compares 3 solutions:
Naïve (brute force) - O(n^6) time
The obvious DP solution - O(n^4) time and O(n^3) space
The more clever DP solution based on Kadane's algorithm - O(n^3) time and O(n^2) space
There are sample runs for n = 10 thru n = 70 in increments of 10 with a nice output comparing run time and space requirements.
Code:
public class MaxSubarray2D {
static int LENGTH;
final static int MAX_VAL = 10;
public static void main(String[] args) {
for (int i = 10; i <= 70; i += 10) {
LENGTH = i;
int[][] a = new int[LENGTH][LENGTH];
for (int row = 0; row < LENGTH; row++) {
for (int col = 0; col < LENGTH; col++) {
a[row][col] = (int) (Math.random() * (MAX_VAL + 1));
if (Math.random() > 0.5D) {
a[row][col] = -a[row][col];
}
//System.out.printf("%4d", a[row][col]);
}
//System.out.println();
}
System.out.println("N = " + LENGTH);
System.out.println("-------");
long start, end;
start = System.currentTimeMillis();
naiveSolution(a);
end = System.currentTimeMillis();
System.out.println(" run time: " + (end - start) + " ms no auxiliary space requirements");
start = System.currentTimeMillis();
dynamicProgammingSolution(a);
end = System.currentTimeMillis();
System.out.println(" run time: " + (end - start) + " ms requires auxiliary space for "
+ ((int) Math.pow(LENGTH, 4)) + " integers");
start = System.currentTimeMillis();
kadane2D(a);
end = System.currentTimeMillis();
System.out.println(" run time: " + (end - start) + " ms requires auxiliary space for " +
+ ((int) Math.pow(LENGTH, 2)) + " integers");
System.out.println();
System.out.println();
}
}
// O(N^2) !!!
public static void kadane2D(int[][] a) {
int[][] s = new int[LENGTH + 1][LENGTH]; // [ending row][sum from row zero to ending row] (rows 1-indexed!)
for (int r = 0; r < LENGTH + 1; r++) {
for (int c = 0; c < LENGTH; c++) {
s[r][c] = 0;
}
}
for (int r = 1; r < LENGTH + 1; r++) {
for (int c = 0; c < LENGTH; c++) {
s[r][c] = s[r - 1][c] + a[r - 1][c];
}
}
int maxSum = Integer.MIN_VALUE;
int maxRowStart = -1;
int maxColStart = -1;
int maxRowEnd = -1;
int maxColEnd = -1;
for (int r1 = 1; r1 < LENGTH + 1; r1++) { // rows 1-indexed!
for (int r2 = r1; r2 < LENGTH + 1; r2++) { // rows 1-indexed!
int[] s1 = new int[LENGTH];
for (int c = 0; c < LENGTH; c++) {
s1[c] = s[r2][c] - s[r1 - 1][c];
}
int max = 0;
int c1 = 0;
for (int c = 0; c < LENGTH; c++) {
max = s1[c] + max;
if (max <= 0) {
max = 0;
c1 = c + 1;
}
if (max > maxSum) {
maxSum = max;
maxRowStart = r1 - 1;
maxColStart = c1;
maxRowEnd = r2 - 1;
maxColEnd = c;
}
}
}
}
System.out.print("KADANE SOLUTION | Max sum: " + maxSum);
System.out.print(" Start: (" + maxRowStart + ", " + maxColStart +
") End: (" + maxRowEnd + ", " + maxColEnd + ")");
}
// O(N^4) !!!
public static void dynamicProgammingSolution(int[][] a) {
int[][][][] dynTable = new int[LENGTH][LENGTH][LENGTH + 1][LENGTH + 1]; // [row][col][height][width]
int maxSum = Integer.MIN_VALUE;
int maxRowStart = -1;
int maxColStart = -1;
int maxRowEnd = -1;
int maxColEnd = -1;
for (int r = 0; r < LENGTH; r++) {
for (int c = 0; c < LENGTH; c++) {
for (int h = 0; h < LENGTH + 1; h++) {
for (int w = 0; w < LENGTH + 1; w++) {
dynTable[r][c][h][w] = 0;
}
}
}
}
for (int r = 0; r < LENGTH; r++) {
for (int c = 0; c < LENGTH; c++) {
for (int h = 1; h <= LENGTH - r; h++) {
int rowTotal = 0;
for (int w = 1; w <= LENGTH - c; w++) {
rowTotal += a[r + h - 1][c + w - 1];
dynTable[r][c][h][w] = rowTotal + dynTable[r][c][h - 1][w];
}
}
}
}
for (int r = 0; r < LENGTH; r++) {
for (int c = 0; c < LENGTH; c++) {
for (int h = 0; h < LENGTH + 1; h++) {
for (int w = 0; w < LENGTH + 1; w++) {
if (dynTable[r][c][h][w] > maxSum) {
maxSum = dynTable[r][c][h][w];
maxRowStart = r;
maxColStart = c;
maxRowEnd = r + h - 1;
maxColEnd = c + w - 1;
}
}
}
}
}
System.out.print(" DP SOLUTION | Max sum: " + maxSum);
System.out.print(" Start: (" + maxRowStart + ", " + maxColStart +
") End: (" + maxRowEnd + ", " + maxColEnd + ")");
}
// O(N^6) !!!
public static void naiveSolution(int[][] a) {
int maxSum = Integer.MIN_VALUE;
int maxRowStart = -1;
int maxColStart = -1;
int maxRowEnd = -1;
int maxColEnd = -1;
for (int rowStart = 0; rowStart < LENGTH; rowStart++) {
for (int colStart = 0; colStart < LENGTH; colStart++) {
for (int rowEnd = 0; rowEnd < LENGTH; rowEnd++) {
for (int colEnd = 0; colEnd < LENGTH; colEnd++) {
int sum = 0;
for (int row = rowStart; row <= rowEnd; row++) {
for (int col = colStart; col <= colEnd; col++) {
sum += a[row][col];
}
}
if (sum > maxSum) {
maxSum = sum;
maxRowStart = rowStart;
maxColStart = colStart;
maxRowEnd = rowEnd;
maxColEnd = colEnd;
}
}
}
}
}
System.out.print(" NAIVE SOLUTION | Max sum: " + maxSum);
System.out.print(" Start: (" + maxRowStart + ", " + maxColStart +
") End: (" + maxRowEnd + ", " + maxColEnd + ")");
}
}
Here is a Java version of Ernesto implementation with some modifications:
public int[][] findMaximumSubMatrix(int[][] matrix){
int dim = matrix.length;
//computing the vertical prefix sum for columns
int[][] ps = new int[dim][dim];
for (int i = 0; i < dim; i++) {
for (int j = 0; j < dim; j++) {
if (j == 0) {
ps[j][i] = matrix[j][i];
} else {
ps[j][i] = matrix[j][i] + ps[j - 1][i];
}
}
}
int maxSum = matrix[0][0];
int top = 0, left = 0, bottom = 0, right = 0;
//Auxiliary variables
int[] sum = new int[dim];
int[] pos = new int[dim];
int localMax;
for (int i = 0; i < dim; i++) {
for (int k = i; k < dim; k++) {
// Kadane over all columns with the i..k rows
reset(sum);
reset(pos);
localMax = 0;
//we keep track of the position of the max value over each Kadane's execution
// notice that we do not keep track of the max value, but only its position
sum[0] = ps[k][0] - (i==0 ? 0 : ps[i-1][0]);
for (int j = 1; j < dim; j++) {
if (sum[j-1] > 0){
sum[j] = sum[j-1] + ps[k][j] - (i==0 ? 0 : ps[i-1][j]);
pos[j] = pos[j-1];
}else{
sum[j] = ps[k][j] - (i==0 ? 0 : ps[i-1][j]);
pos[j] = j;
}
if (sum[j] > sum[localMax]){
localMax = j;
}
}//Kadane ends here
if (sum[localMax] > maxSum){
/* sum[localMax] is the new max value
the corresponding submatrix goes from rows i..k.
and from columns pos[localMax]..localMax
*/
maxSum = sum[localMax];
top = i;
left = pos[localMax];
bottom = k;
right = localMax;
}
}
}
System.out.println("Max SubMatrix determinant = " + maxSum);
//composing the required matrix
int[][] output = new int[bottom - top + 1][right - left + 1];
for(int i = top, k = 0; i <= bottom; i++, k++){
for(int j = left, l = 0; j <= right ; j++, l++){
output[k][l] = matrix[i][j];
}
}
return output;
}
private void reset(int[] a) {
for (int index = 0; index < a.length; index++) {
a[index] = 0;
}
}
With the help of the Algorithmist and Larry and a modification of Kadane's Algorithm, here is my solution:
int dim = matrix.length;
//computing the vertical prefix sum for columns
int[][] ps = new int[dim][dim];
for (int i = 0; i < dim; i++) {
for (int j = 0; j < dim; j++) {
if (j == 0) {
ps[j][i] = matrix[j][i];
} else {
ps[j][i] = matrix[j][i] + ps[j - 1][i];
}
}
}
int maxSoFar = 0;
int min , subMatrix;
//iterate over the possible combinations applying Kadane's Alg.
for (int i = 0; i < dim; i++) {
for (int j = i; j < dim; j++) {
min = 0;
subMatrix = 0;
for (int k = 0; k < dim; k++) {
if (i == 0) {
subMatrix += ps[j][k];
} else {
subMatrix += ps[j][k] - ps[i - 1 ][k];
}
if(subMatrix < min){
min = subMatrix;
}
if((subMatrix - min) > maxSoFar){
maxSoFar = subMatrix - min;
}
}
}
}
The only thing left is to determine the submatrix elements, i.e: the top left and the bottom right corner of the submatrix. Anyone suggestion?
this is my implementation of 2D Kadane algorithm. I think it is more clear. The concept is based on just kadane algorithm. The first and second loop of the main part (that is in the bottom of the code) is to pick every combination of the rows and 3rd loop is to use 1D kadane algorithm by every following column sum (that can be computed in const time because of preprocessing of matrix by subtracting values from two picked (from combintation) rows). Here is the code:
int [][] m = {
{1,-5,-5},
{1,3,-5},
{1,3,-5}
};
int N = m.length;
// summing columns to be able to count sum between two rows in some column in const time
for (int i=0; i<N; ++i)
m[0][i] = m[0][i];
for (int j=1; j<N; ++j)
for (int i=0; i<N; ++i)
m[j][i] = m[j][i] + m[j-1][i];
int total_max = 0, sum;
for (int i=0; i<N; ++i) {
for (int k=i; k<N; ++k) { //for each combination of rows
sum = 0;
for (int j=0; j<N; j++) { //kadane algorithm for every column
sum += i==0 ? m[k][j] : m[k][j] - m[i-1][j]; //for first upper row is exception
total_max = Math.max(sum, total_max);
}
}
}
System.out.println(total_max);
I am going to post an answer here and can add actual c++ code if it is requested because I had recently worked through this. Some rumors of a divide and conqueror that can solve this in O(N^2) are out there but I haven't seen any code to support this. In my experience the following is what I have found.
O(i^3j^3) -- naive brute force method
o(i^2j^2) -- dynamic programming with memoization
O(i^2j) -- using max contiguous sub sequence for an array
if ( i == j )
O(n^6) -- naive
O(n^4) -- dynamic programming
O(n^3) -- max contiguous sub sequence
Have a look at JAMA package; I believe it will make your life easier.
Here is the C# solution. Ref: http://www.algorithmist.com/index.php/UVa_108
public static MaxSumMatrix FindMaxSumSubmatrix(int[,] inMtrx)
{
MaxSumMatrix maxSumMtrx = new MaxSumMatrix();
// Step 1. Create SumMatrix - do the cumulative columnar summation
// S[i,j] = S[i-1,j]+ inMtrx[i-1,j];
int m = inMtrx.GetUpperBound(0) + 2;
int n = inMtrx.GetUpperBound(1)+1;
int[,] sumMatrix = new int[m, n];
for (int i = 1; i < m; i++)
{
for (int j = 0; j < n; j++)
{
sumMatrix[i, j] = sumMatrix[i - 1, j] + inMtrx[i - 1, j];
}
}
PrintMatrix(sumMatrix);
// Step 2. Create rowSpans starting each rowIdx. For these row spans, create a 1-D array r_ij
for (int x = 0; x < n; x++)
{
for (int y = x; y < n; y++)
{
int[] r_ij = new int[n];
for (int k = 0; k < n; k++)
{
r_ij[k] = sumMatrix[y + 1,k] - sumMatrix[x, k];
}
// Step 3. Find MaxSubarray of this r_ij. If the sum is greater than the last recorded sum =>
// capture Sum, colStartIdx, ColEndIdx.
// capture current x as rowTopIdx, y as rowBottomIdx.
MaxSum currMaxSum = KadanesAlgo.FindMaxSumSubarray(r_ij);
if (currMaxSum.maxSum > maxSumMtrx.sum)
{
maxSumMtrx.sum = currMaxSum.maxSum;
maxSumMtrx.colStart = currMaxSum.maxStartIdx;
maxSumMtrx.colEnd = currMaxSum.maxEndIdx;
maxSumMtrx.rowStart = x;
maxSumMtrx.rowEnd = y;
}
}
}
return maxSumMtrx;
}
public static void PrintMatrix(int[,] matrix)
{
int endRow = matrix.GetUpperBound(0);
int endCol = matrix.GetUpperBound(1);
PrintMatrix(matrix, 0, endRow, 0, endCol);
}
public static void PrintMatrix(int[,] matrix, int startRow, int endRow, int startCol, int endCol)
{
StringBuilder sb = new StringBuilder();
for (int i = startRow; i <= endRow; i++)
{
sb.Append(Environment.NewLine);
for (int j = startCol; j <= endCol; j++)
{
sb.Append(string.Format("{0} ", matrix[i,j]));
}
}
Console.WriteLine(sb.ToString());
}
// Given an NxN matrix of positive and negative integers, write code to find the sub-matrix with the largest possible sum
public static MaxSum FindMaxSumSubarray(int[] inArr)
{
int currMax = 0;
int currStartIndex = 0;
// initialize maxSum to -infinity, maxStart and maxEnd idx to 0.
MaxSum mx = new MaxSum(int.MinValue, 0, 0);
// travers through the array
for (int currEndIndex = 0; currEndIndex < inArr.Length; currEndIndex++)
{
// add element value to the current max.
currMax += inArr[currEndIndex];
// if current max is more that the last maxSum calculated, set the maxSum and its idx
if (currMax > mx.maxSum)
{
mx.maxSum = currMax;
mx.maxStartIdx = currStartIndex;
mx.maxEndIdx = currEndIndex;
}
if (currMax < 0) // if currMax is -ve, change it back to 0
{
currMax = 0;
currStartIndex = currEndIndex + 1;
}
}
return mx;
}
struct MaxSum
{
public int maxSum;
public int maxStartIdx;
public int maxEndIdx;
public MaxSum(int mxSum, int mxStart, int mxEnd)
{
this.maxSum = mxSum;
this.maxStartIdx = mxStart;
this.maxEndIdx = mxEnd;
}
}
class MaxSumMatrix
{
public int sum = int.MinValue;
public int rowStart = -1;
public int rowEnd = -1;
public int colStart = -1;
public int colEnd = -1;
}
Here is my solution. It's O(n^3) in time and O(n^2) space.
https://gist.github.com/toliuweijing/6097144
// 0th O(n) on all candidate bottoms #B.
// 1th O(n) on candidate tops #T.
// 2th O(n) on finding the maximum #left/#right match.
int maxRect(vector<vector<int> >& mat) {
int n = mat.size();
vector<vector<int> >& colSum = mat;
for (int i = 1 ; i < n ; ++i)
for (int j = 0 ; j < n ; ++j)
colSum[i][j] += colSum[i-1][j];
int optrect = 0;
for (int b = 0 ; b < n ; ++b) {
for (int t = 0 ; t <= b ; ++t) {
int minLeft = 0;
int rowSum[n];
for (int i = 0 ; i < n ; ++i) {
int col = t == 0 ? colSum[b][i] : colSum[b][i] - colSum[t-1][i];
rowSum[i] = i == 0? col : col + rowSum[i-1];
optrect = max(optrect, rowSum[i] - minLeft);
minLeft = min(minLeft, rowSum[i]);
}
}
}
return optrect;
}
I would just parse the NxN array removing the -ves whatever remains is the highest sum of a sub matrix.
The question doesn't say you have to leave the original matrix intact or that the order matters.