# 1st to Kth shortest path lengths from node 1 to N in given Graph

Given a directed and weighted graph of **N **nodes and **M** edges, the task is to find the **1st to K ^{th}**

**shortest path**lengths from node

**1**to

**N**.

**Examples:**

Input:N = 4, M = 6, K = 3, edges = {{1, 2, 1}, {1, 3, 3}, {2, 3, 2}, {2, 4, 6}, {3, 2, 8}, {3, 4, 1}}Output:4 4 7Explanation:The shortest path length from 1 to N is 4, 2nd shortest length is also 4 and 3rd shortest length is 7.

Input:N = 3, M = 3, K = 2, edges = {{1, 2, 2}, {2, 3, 2}, {1, 3, 1}}Output:1 4

**Approach: **The idea is to traverse all vertices of the graph using BFS and use priority queue to store the vertices for which the shortest distance is not finalized yet, and also to get the **minimum** distance vertex. Follow the steps below for the approach.

- Initialize a priority queue, say,
**pq**of size**N**to store the vertex number and distance value. - Initialize a 2-d vector, say,
**dis**of size**N*K**and initialize all values with a very large number, say, 1e9. - Set
**dis[1][0]**to**zero,**the distance value of the source vertex. - Iterate while
**pq**is not empty.- Pop the value from
**pq,**and store the vertex value in variable**u**and vertex distance in variable**d.** - If
**d**is greater than the distance**u**, then**continue.** - For every adjacent vertex
**v**of**u**check if the**Kth**distance value is more than the weight of**u-v**plus the distance value of**u,**then update the**Kth**distance value of**v**and sort the**k**distances of vertex**v.**

- Pop the value from

Below is the implementation of the above approach:

## C++14

`// C++ implementation of above approach` `#include <bits/stdc++.h>` `using` `namespace` `std;` ` ` `// Function to find K shortest path lengths` `void` `findKShortest(` `int` `edges[][3], ` `int` `n, ` `int` `m, ` `int` `k)` `{` ` ` ` ` `// Initialize graph` ` ` `vector<vector<pair<` `int` `, ` `int` `> > > g(n + 1);` ` ` `for` `(` `int` `i = 0; i < m; i++) {` ` ` `// Storing edges` ` ` `g[edges[i][0]].push_back({ edges[i][1], edges[i][2] });` ` ` `}` ` ` ` ` `// Vector to store distances` ` ` `vector<vector<` `int` `> > dis(n + 1, vector<` `int` `>(k, 1e9));` ` ` ` ` `// Initialization of priority queue` ` ` `priority_queue<pair<` `int` `, ` `int` `>,` ` ` `vector<pair<` `int` `, ` `int` `> >,` ` ` `greater<pair<` `int` `, ` `int` `> > >` ` ` `pq;` ` ` `pq.push({ 0, 1 });` ` ` `dis[1][0] = 0;` ` ` ` ` `// while pq has elements` ` ` `while` `(!pq.empty()) {` ` ` `// Storing the node value` ` ` `int` `u = pq.top().second;` ` ` ` ` `// Storing the distance value` ` ` `int` `d = (pq.top().first);` ` ` `pq.pop();` ` ` `if` `(dis[u][k - 1] < d)` ` ` `continue` `;` ` ` `vector<pair<` `int` `, ` `int` `> > v = g[u];` ` ` ` ` `// Traversing the adjacency list` ` ` `for` `(` `int` `i = 0; i < v.size(); i++) {` ` ` `int` `dest = v[i].first;` ` ` `int` `cost = v[i].second;` ` ` ` ` `// Checking for the cost` ` ` `if` `(d + cost < dis[dest][k - 1]) {` ` ` `dis[dest][k - 1] = d + cost;` ` ` ` ` `// Sorting the distances` ` ` `sort(dis[dest].begin(), dis[dest].end());` ` ` ` ` `// Pushing elements to priority queue` ` ` `pq.push({ (d + cost), dest });` ` ` `}` ` ` `}` ` ` `}` ` ` ` ` `// Printing K shortest paths` ` ` `for` `(` `int` `i = 0; i < k; i++) {` ` ` `cout << dis[n][i] << ` `" "` `;` ` ` `}` `}` ` ` `// Driver Code` `int` `main()` `{` ` ` ` ` `// Given Input` ` ` `int` `N = 4, M = 6, K = 3;` ` ` `int` `edges[][3] = { { 1, 2, 1 }, { 1, 3, 3 },` ` ` `{ 2, 3, 2 }, { 2, 4, 6 },` ` ` `{ 3, 2, 8 }, { 3, 4, 1 } };` ` ` ` ` `// Function Call` ` ` `findKShortest(edges, N, M, K);` ` ` ` ` `return` `0;` `}` |

**Output**

4 4 7

**Time Complexity:** O((N+M)*KlogK)**Auxiliary Space: **O(NK)

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