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Shortest Paths with Negative Edge Weights

Authors: Benjamin Qi, Andi Qu

Returning to Bellman-Ford and Floyd-Warshall.

Bellman-Ford

Shortest Paths

Focus Problem – read through this problem before continuing!

Solution

This section is not complete.

Any help would be appreciated! Just submit a Pull Request on Github.

Finding Negative Cycles

Focus Problem – read through this problem before continuing!

Solution

This section is not complete.

Any help would be appreciated! Just submit a Pull Request on Github.

Simple Linear Programming

You can also use shortest path algorithms to solve the following problem (a very simple linear program).

Given variables with constraints in the form , compute a feasible solution.

Resources

Resources
MITLinear Programming Trick

Problems

Timeline (USACO Camp):

  • equivalent to Timeline (Gold) except and negative values of are possible.
StatusSourceProblem NameDifficultyTagsSolutionURL
RMINormalShow Sketch

Floyd-Warshall

Focus Problem – read through this problem before continuing!

Solution - APSP

C++

const int MOD = 1000000007;
const ll INF = 1e18;
int n,m,q;
ll dist[150][150], bad[150][150];
void solve() {
F0R(i,n) F0R(j,n) dist[i][j] = INF, bad[i][j] = 0;
F0R(i,n) dist[i][i] = 0;
F0R(i,m) {

Problems

StatusSourceProblem NameDifficultyTagsSolutionURL
APIOHard
Show Tags

APSP, Binary Search

Modified Dijkstra

The Dijkstra code presented earlier will still give correct results if there are no negative cycles. However, the same running time bound no longer applies, as demonstrated by subtasks 1-6 of the following problem.

StatusSourceProblem NameDifficultyTagsSolutionURL
APIOHard
Show Tags

SP, Output-only

This problem forces you to analyze the inner workings of the three shortest-path algorithms we presented here. It also teaches you about how problemsetters could create hack cases!

Module Progress:

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