There is an integer array d which does not contain more than two elements of the same value. How many distinct ascending triples (d[i] < d[j] < d[k], i < j < k) are present?
Input format:
The first line contains an integer N denoting the number of elements in the array. This is followed by a single line containing N integers separated by a single space with no leading/trailing spaces
Output format:
A single integer that denotes the number of distinct ascending triples present in the array
Constraints:
N <= 10^5
Every value in the array is present at most twice
Every value in the array is a 32-bit positive integer
Sample input:
6
1 1 2 2 3 4
Sample output:
4
Explanation:
The distinct triplets are
(1,2,3)
(1,2,4)
(1,3,4)
(2,3,4)
Another test case:
Input:
10
1 1 5 4 3 6 6 5 9 10
Output:
28
I tried to solve using DP. But out of 15 test cases only 7 test cases passed.
Please help solve this problem.
You should note that you only need to know the number of elements that are smaller/larger than a particular element to know how many triples it serves as the middle point for. Using this you can calculate the number of triples quite easily, the only remaining problem is to get rid of duplicates, but given that you are limited to at most 2 of the same element, this is trivial.
I solved using a Binary Index Tree http://community.topcoder.com/tc?module=Static&d1=tutorials&d2=binaryIndexedTrees.
I also did a small write up, http://www.kesannmcclean.com/?p=223.
package com.jai;
import java.io.BufferedReader;
import java.io.InputStreamReader;
import java.util.Arrays;
import java.util.HashMap;
public class Triplets {
int[] lSmaller, rLarger, treeArray, dscArray, lFlags, rFlags;
int size, count = 0;
Triplets(int aSize, int[] inputArray) {
size = aSize;
lSmaller = new int[size];
rLarger = new int[size];
dscArray = new int[size];
int[] tmpArray = Arrays.copyOf(inputArray, inputArray.length);
Arrays.sort(tmpArray);
HashMap<Integer, Integer> tmpMap = new HashMap<Integer, Integer>(size);
for (int i = 0; i < size; i++) {
if (!tmpMap.containsKey(tmpArray[i])) {
count++;
tmpMap.put(tmpArray[i], count);
}
}
count++;
treeArray = new int[count];
lFlags = new int[count];
rFlags = new int[count];
for (int i = 0; i < size; i++) {
dscArray[i] = tmpMap.get(inputArray[i]);
}
}
void update(int idx) {
while (idx < count) {
treeArray[idx]++;
idx += (idx & -idx);
}
}
int read(int index) {
int sum = 0;
while (index > 0) {
sum += treeArray[index];
index -= (index & -index);
}
return sum;
}
void countLeftSmaller() {
Arrays.fill(treeArray, 0);
Arrays.fill(lSmaller, 0);
Arrays.fill(lFlags, 0);
for (int i = 0; i < size; i++) {
int val = dscArray[i];
lSmaller[i] = read(val - 1);
if (lFlags[val] == 0) {
update(val);
lFlags[val] = i + 1;
} else {
lSmaller[i] -= lSmaller[lFlags[val] - 1];
}
}
}
void countRightLarger() {
Arrays.fill(treeArray, 0);
Arrays.fill(rLarger, 0);
Arrays.fill(rFlags, 0);
for (int i = size - 1; i >= 0; i--) {
int val = dscArray[i];
rLarger[i] = read(count - 1) - read(val);
if (rFlags[val] == 0) {
update(val);
rFlags[val] = i + 1;
}
}
}
long countTriplets() {
long sum = 0;
for (int i = 0; i < size; i++) {
sum += lSmaller[i] * rLarger[i];
}
return sum;
}
public static void main(String[] args) throws Exception {
BufferedReader br = new BufferedReader(new InputStreamReader(System.in));
int N = Integer.parseInt(br.readLine());
int[] a = new int[N];
String[] strs = br.readLine().split(" ");
for (int i = 0; i < N; i++)
a[i] = Integer.parseInt(strs[i]);
Triplets sol = new Triplets(N, a);
sol.countLeftSmaller();
sol.countRightLarger();
System.out.println(sol.countTriplets());
}
}
For tentative algorithm that I came up with, it should be:
(K-1)!^2
where K is number of unique elements.
EDIT
After more thinking about this:
SUM[i=1,K-2] SUM[j=i+1,K-1] SUM[m=j+1,K] 1
=> SUM[i=1,K-2] (SUM[j=i+1,K-1] (K-j))
if the input is not sorted (the question is not clear about this): sort it
remove the duplicated items (this step could be conbined with the first step)
now pick 3 items. Since the items are already sorted, the three chosen items are ordered as well
IIRC there are (n!) / ((n-3)! * 3!) ways to pick the three items; with n := the number of unique items
#hadron: exactly, I couldn get my head around on why it should be 28 and not 35 for a set of 7 distinct numbers *
[Since the ques is about ascending triplets, repeated numbers can be discarded].
btw, here's a very bad Java solution(N^3):
I have also printed out the possible triplets:
I'm also thinking about some function that dictates the no: of triplets possible for input 'N'
4 4
5 10
6 20
7 35
8 56
9 84
package org.HackerRank.AlgoChallenges;
import java.util.Iterator;
import java.util.Scanner;
import java.util.TreeSet;
public class Triplets {
public static void main(String[] args) {
Scanner scanner = new Scanner(System.in);
int result = 0;
int n = scanner.nextInt();
Object[] array = new Object[n];
TreeSet<Integer> treeSet = new TreeSet<Integer>();
/*
* for (int i = 0; i < n; i++) { array[i] = scanner.nextInt(); }
*/
while (n>0) {
treeSet.add(scanner.nextInt());
n--;
}
scanner.close();
Iterator<Integer> iterator = treeSet.iterator();
int i =0;
while (iterator.hasNext()) {
//System.out.println("TreeSet["+i+"] : "+iterator.next());
array[i] = iterator.next();
//System.out.println("Array["+i+"] : "+array[i]);
i++;
}
for (int j = 0; j < (array.length-2); j++) {
for (int j2 = (j+1); j2 < array.length-1; j2++) {
for (int k = (j2+1); k < array.length; k++) {
if(array[j]!=null && array[j2]!=null && array[k]!=null){
System.out.println("{ "+array[j]+", "+array[j2]+", "+array[k]+" }");
result++;
}
}
}
}
System.out.println(result);
}
One of solution in python:
from itertools import combinations as comb
def triplet(lis):
done = dict()
result = set()
for ind, num in enumerate(lis):
if num not in done:
index = ind+1
for elm in comb(lis[index:], 2):
s,t = elm[0], elm[1]
if (num < s < t):
done.setdefault(num, None)
fin = (num,s,t)
if fin not in result:
result.add(fin)
return len(result)
test = int(raw_input())
lis = [int(_) for _ in raw_input().split()]
print triplet(lis)
Do you care about complexity?
Is the input array sorted?
if you don't mind about complexity you can solve it in complexity of N^3.
The solution with complexity N^3:
If it not sorted, then sorted the array.
Use 3 for loops one inside the other and go threw the array 3 times for each number.
Use hash map to count all the triples. The key will be the triple it self and the value will be the number of occurences.
It should be something like this:
for (i1=0; i1<N; i1++) {
for (i2=i1; i2<N; i2++) {
for (i3=i2; i3<N; i3++) {
if (N[i1] < N[i2] < N[i3]) {
/* if the triple exists in the hash then
add 1 to its value
else
put new triple to the hash with
value 1
*/
}
}
}
}
Result = number of triples in the hash;
I didn't try it but I think it should work.
We need to find pair of numbers in an array whose sum is equal to a given value.
A = {6,4,5,7,9,1,2}
Sum = 10
Then the pairs are - {6,4} , {9,1}
I have two solutions for this .
an O(nlogn) solution - sort + check sum with 2 iterators (beginning and end).
an O(n) solution - hashing the array. Then checking if sum-hash[i] exists in the hash table or not.
But , the problem is that although the second solution is O(n) time , but uses O(n) space as well.
So , I was wondering if we could do it in O(n) time and O(1) space. And this is NOT homework!
Use in-place radix sort and OP's first solution with 2 iterators, coming towards each other.
If numbers in the array are not some sort of multi-precision numbers and are, for example, 32-bit integers, you can sort them in 2*32 passes using practically no additional space (1 bit per pass). Or 2*8 passes and 16 integer counters (4 bits per pass).
Details for the 2 iterators solution:
First iterator initially points to first element of the sorted array and advances forward. Second iterator initially points to last element of the array and advances backward.
If sum of elements, referenced by iterators, is less than the required value, advance first iterator. If it is greater than the required value, advance second iterator. If it is equal to the required value, success.
Only one pass is needed, so time complexity is O(n). Space complexity is O(1). If radix sort is used, complexities of the whole algorithm are the same.
If you are interested in related problems (with sum of more than 2 numbers), see "Sum-subset with a fixed subset size" and "Finding three elements in an array whose sum is closest to an given number".
This is a classic interview question from Microsoft research Asia.
How to Find 2 numbers in an unsorted array equal to a given sum.
[1]brute force solution
This algorithm is very simple. The time complexity is O(N^2)
[2]Using binary search
Using bianry searching to find the Sum-arr[i] with every arr[i], The time complexity can be reduced to O(N*logN)
[3]Using Hash
Base on [2] algorithm and use hash, the time complexity can be reduced to O(N), but this solution will add the O(N) space of hash.
[4]Optimal algorithm:
Pseduo-code:
for(i=0;j=n-1;i<j)
if(arr[i]+arr[j]==sum) return (i,j);
else if(arr[i]+arr[j]<sum) i++;
else j--;
return(-1,-1);
or
If a[M] + a[m] > I then M--
If a[M] + a[m] < I then m++
If a[M] + a[m] == I you have found it
If m > M, no such numbers exist.
And, Is this quesiton completely solved? No. If the number is N. This problem will become very complex.
The quesiton then:
How can I find all the combination cases with a given number?
This is a classic NP-Complete problem which is called subset-sum.
To understand NP/NPC/NP-Hard you'd better to read some professional books.
References:
[1]http://www.quora.com/Mathematics/How-can-I-find-all-the-combination-cases-with-a-given-number
[2]http://en.wikipedia.org/wiki/Subset_sum_problem
for (int i=0; i < array.size(); i++){
int value = array[i];
int diff = sum - value;
if (! hashSet.contains(diffvalue)){
hashSet.put(value,value);
} else{
printf(sum = diffvalue + hashSet.get(diffvalue));
}
}
--------
Sum being sum of 2 numbers.
public void printPairsOfNumbers(int[] a, int sum){
//O(n2)
for (int i = 0; i < a.length; i++) {
for (int j = i+1; j < a.length; j++) {
if(sum - a[i] == a[j]){
//match..
System.out.println(a[i]+","+a[j]);
}
}
}
//O(n) time and O(n) space
Set<Integer> cache = new HashSet<Integer>();
cache.add(a[0]);
for (int i = 1; i < a.length; i++) {
if(cache.contains(sum - a[i])){
//match//
System.out.println(a[i]+","+(sum-a[i]));
}else{
cache.add(a[i]);
}
}
}
Create a dictionary with pairs Key (number from the list) and the Value is the number which is necessary to obtain a desired value. Next, check the presence of the pairs of numbers in the list.
def check_sum_in_list(p_list, p_check_sum):
l_dict = {i: (p_check_sum - i) for i in p_list}
for key, value in l_dict.items():
if key in p_list and value in p_list:
return True
return False
if __name__ == '__main__':
l1 = [1, 3, 7, 12, 72, 2, 8]
l2 = [1, 2, 2, 4, 7, 4, 13, 32]
print(check_sum_in_list(l1, 10))
print(check_sum_in_list(l2, 99))
Output:
True
Flase
version 2
import random
def check_sum_in_list(p_list, p_searched_sum):
print(list(p_list))
l_dict = {i: p_searched_sum - i for i in set(p_list)}
for key, value in l_dict.items():
if key in p_list and value in p_list:
if p_list.index(key) != p_list.index(value):
print(key, value)
return True
return False
if __name__ == '__main__':
l1 = []
for i in range(1, 2000000):
l1.append(random.randrange(1, 1000))
j = 0
i = 9
while i < len(l1):
if check_sum_in_list(l1[j:i], 100):
print('Found')
break
else:
print('Continue searching')
j = i
i = i + 10
Output:
...
[154, 596, 758, 924, 797, 379, 731, 278, 992, 167]
Continue searching
[808, 730, 216, 15, 261, 149, 65, 386, 670, 770]
Continue searching
[961, 632, 39, 888, 61, 18, 166, 167, 474, 108]
39 61
Finded
[Finished in 3.9s]
If you assume that the value M to which the pairs are suppose to sum is constant and that the entries in the array are positive, then you can do this in one pass (O(n) time) using M/2 pointers (O(1) space) as follows. The pointers are labeled P1,P2,...,Pk where k=floor(M/2). Then do something like this
for (int i=0; i<N; ++i) {
int j = array[i];
if (j < M/2) {
if (Pj == 0)
Pj = -(i+1); // found smaller unpaired
else if (Pj > 0)
print(Pj-1,i); // found a pair
Pj = 0;
} else
if (Pj == 0)
Pj = (i+1); // found larger unpaired
else if (Pj < 0)
print(Pj-1,i); // found a pair
Pj = 0;
}
}
You can handle repeated entries (e.g. two 6's) by storing the indices as digits in base N, for example. For M/2, you can add the conditional
if (j == M/2) {
if (Pj == 0)
Pj = i+1; // found unpaired middle
else
print(Pj-1,i); // found a pair
Pj = 0;
}
But now you have the problem of putting the pairs together.
Does the obvious solution not work (iterating over every consecutive pair) or are the two numbers in any order?
In that case, you could sort the list of numbers and use random sampling to partition the sorted list until you have a sublist that is small enough to be iterated over.
public static ArrayList<Integer> find(int[] A , int target){
HashSet<Integer> set = new HashSet<Integer>();
ArrayList<Integer> list = new ArrayList<Integer>();
int diffrence = 0;
for(Integer i : A){
set.add(i);
}
for(int i = 0; i <A.length; i++){
diffrence = target- A[i];
if(set.contains(diffrence)&&A[i]!=diffrence){
list.add(A[i]);
list.add(diffrence);
return list;
}
}
return null;
}
`package algorithmsDesignAnalysis;
public class USELESStemp {
public static void main(String[] args){
int A[] = {6, 8, 7, 5, 3, 11, 10};
int sum = 12;
int[] B = new int[A.length];
int Max =A.length;
for(int i=0; i<A.length; i++){
B[i] = sum - A[i];
if(B[i] > Max)
Max = B[i];
if(A[i] > Max)
Max = A[i];
System.out.print(" " + B[i] + "");
} // O(n) here;
System.out.println("\n Max = " + Max);
int[] Array = new int[Max+1];
for(int i=0; i<B.length; i++){
Array[B[i]] = B[i];
} // O(n) here;
for(int i=0; i<A.length; i++){
if (Array[A[i]] >= 0)
System.out.println("We got one: " + A[i] +" and " + (sum-A[i]));
} // O(n) here;
} // end main();
/******
Running time: 3*O(n)
*******/
}
Below code takes the array and the number N as the target sum.
First the array is sorted, then a new array containing the
remaining elements are taken and then scanned not by binary search
but simple scanning of the remainder and the array simultaneously.
public static int solution(int[] a, int N) {
quickSort(a, 0, a.length-1); // nlog(n)
int[] remainders = new int[a.length];
for (int i=0; i<a.length; i++) {
remainders[a.length-1-i] = N - a[i]; // n
}
int previous = 0;
for (int j=0; j<a.length; j++) { // ~~ n
int k = previous;
while(k < remainders.length && remainders[k] < a[j]) {
k++;
}
if(k < remainders.length && remainders[k] == a[j]) {
return 1;
}
previous = k;
}
return 0;
}
Shouldn't iterating from both ends just solve the problem?
Sort the array. And start comparing from both ends.
if((arr[start] + arr[end]) < sum) start++;
if((arr[start] + arr[end]) > sum) end--;
if((arr[start] + arr[end]) = sum) {print arr[start] "," arr[end] ; start++}
if(start > end) break;
Time Complexity O(nlogn)
if its a sorted array and we need only pair of numbers and not all the pairs we can do it like this:
public void sums(int a[] , int x){ // A = 1,2,3,9,11,20 x=11
int i=0 , j=a.length-1;
while(i < j){
if(a[i] + a[j] == x) system.out.println("the numbers : "a[x] + " " + a[y]);
else if(a[i] + a[j] < x) i++;
else j--;
}
}
1 2 3 9 11 20 || i=0 , j=5 sum=21 x=11
1 2 3 9 11 20 || i=0 , j=4 sum=13 x=11
1 2 3 9 11 20 || i=0 , j=4 sum=11 x=11
END
The following code returns true if two integers in an array match a compared integer.
function compareArraySums(array, compare){
var candidates = [];
function compareAdditions(element, index, array){
if(element <= y){
candidates.push(element);
}
}
array.forEach(compareAdditions);
for(var i = 0; i < candidates.length; i++){
for(var j = 0; j < candidates.length; j++){
if (i + j === y){
return true;
}
}
}
}
Python 2.7 Implementation:
import itertools
list = [1, 1, 2, 3, 4, 5,]
uniquelist = set(list)
targetsum = 5
for n in itertools.combinations(uniquelist, 2):
if n[0] + n[1] == targetsum:
print str(n[0]) + " + " + str(n[1])
Output:
1 + 4
2 + 3
https://github.com/clockzhong/findSumPairNumber
#! /usr/bin/env python
import sys
import os
import re
#get the number list
numberListStr=raw_input("Please input your number list (seperated by spaces)...\n")
numberList=[int(i) for i in numberListStr.split()]
print 'you have input the following number list:'
print numberList
#get the sum target value
sumTargetStr=raw_input("Please input your target number:\n")
sumTarget=int(sumTargetStr)
print 'your target is: '
print sumTarget
def generatePairsWith2IndexLists(list1, list2):
result=[]
for item1 in list1:
for item2 in list2:
#result.append([item1, item2])
result.append([item1+1, item2+1])
#print result
return result
def generatePairsWithOneIndexLists(list1):
result=[]
index = 0
while index< (len(list1)-1):
index2=index+1
while index2 < len(list1):
#result.append([list1[index],list1[index2]])
result.append([list1[index]+1,list1[index2]+1])
index2+=1
index+=1
return result
def getPairs(numList, target):
pairList=[]
candidateSlots=[] ##we have (target-1) slots
#init the candidateSlots list
index=0
while index < target+1:
candidateSlots.append(None)
index+=1
#generate the candidateSlots, contribute O(n) complexity
index=0
while index<len(numList):
if numList[index]<=target and numList[index]>=0:
#print 'index:',index
#print 'numList[index]:',numList[index]
#print 'len(candidateSlots):',len(candidateSlots)
if candidateSlots[numList[index]]==None:
candidateSlots[numList[index]]=[index]
else:
candidateSlots[numList[index]].append(index)
index+=1
#print candidateSlots
#generate the pairs list based on the candidateSlots[] we just created
#contribute O(target) complexity
index=0
while index<=(target/2):
if candidateSlots[index]!=None and candidateSlots[target-index]!=None:
if index!=(target-index):
newPairList=generatePairsWith2IndexLists(candidateSlots[index], candidateSlots[target-index])
else:
newPairList=generatePairsWithOneIndexLists(candidateSlots[index])
pairList+=newPairList
index+=1
return pairList
print getPairs(numberList, sumTarget)
I've successfully implemented one solution with Python under O(n+m) time and space cost.
The "m" means the target value which those two numbers' sum need equal to.
I believe this is the lowest cost could get. Erict2k used itertools.combinations, it'll also cost similar or higher time&space cost comparing my algorithm.
If numbers aren't very big, you can use fast fourier transform to multiply two polynomials and then in O(1) check if coefficient before x^(needed sum) sum is more than zero. O(n log n) total!
// Java implementation using Hashing
import java.io.*;
class PairSum
{
private static final int MAX = 100000; // Max size of Hashmap
static void printpairs(int arr[],int sum)
{
// Declares and initializes the whole array as false
boolean[] binmap = new boolean[MAX];
for (int i=0; i<arr.length; ++i)
{
int temp = sum-arr[i];
// checking for condition
if (temp>=0 && binmap[temp])
{
System.out.println("Pair with given sum " +
sum + " is (" + arr[i] +
", "+temp+")");
}
binmap[arr[i]] = true;
}
}
// Main to test the above function
public static void main (String[] args)
{
int A[] = {1, 4, 45, 6, 10, 8};
int n = 16;
printpairs(A, n);
}
}
public static void Main(string[] args)
{
int[] myArray = {1,2,3,4,5,6,1,4,2,2,7 };
int Sum = 9;
for (int j = 1; j < myArray.Length; j++)
{
if (myArray[j-1]+myArray[j]==Sum)
{
Console.WriteLine("{0}, {1}",myArray[j-1],myArray[j]);
}
}
Console.ReadLine();
}