Effective Problem Solving Strategies for Competitive Programming

1. Competition Programming Analyzing and Solving problems Christopher Moh 2005

2. Steps in solving a problem • Reading the problem • Formulation of a solution • Implementation the solution • Testing the solution Christopher Moh 2005

3. Reading the problem • Read the WHOLE problem! • May have subtle details in problem descriptions that are essential to solve the problem efficiently • Examining Time and Space constraints • Watching out for obvious properties Christopher Moh 2005

4. Time and Space Constraints • Time Constraints usually are more important than Space Constraints • What kind of algorithm works? • Good average case? • Good worst case? • Gives an idea of the kind of algorithm that solves the problem • E.g. Huge graph problems usually imply some linear-time traversal solution Christopher Moh 2005

5. Time and Space Constraints • Does the problem specify the size of the test cases? • If there are only one or two test cases that are huge compared to the rest… • DON’T SOLVE FOR THESE TEST CASES IF YOU DON’T KNOW THE ADVANCED ALGORITHM • There usually will be a basic algorithm that will solve the rest with suitable optimizations and data structures Christopher Moh 2005

6. Properties I • What unique properties are explicitly stated or easily implied? • Unique Numbers? • At most one path between any two nodes implies a forest of trees • Exactly one path between any two nodes implies a tree • No return path may imply a DAG • Classification in a system of “black” and “white” etc may imply a bipartite graph Christopher Moh 2005

7. Formulation • How can I solve this problem? • As a Graph problem? • As a DP problem? • Sorting problem? • Clever data structures? • Using unique properties of the problem? • Brute Force? Christopher Moh 2005

8. Breaking down problems • Can this problem be broken down into a decent number of sub-problems that can be combined to solve the problem? • Is there an inherent, or maybe a subtle implied order that we can use? • Can I use a solution to a similar problem of a smaller dimension? Christopher Moh 2005

9. Properties II • Are there number properties I can use? • Is this a special graph with unique properties e.g. trees, DAGs, etc.? • Is sorting allowed and what advantage can sorting give me? • Does a greedy solution work? Christopher Moh 2005

10. Implementation • Data Structures • Can I get away with using simple data structures and still beat the time limit? • If not, what works? • Heaps? • Interval trees? • Cumulative Sum data structures? [ See IOI 2001 Day 1 Q1 ] Christopher Moh 2005

11. Implementation • Dynamic Programming solution • What do I have to keep track of? • How can I evaluate the recurrence relation? How should my loops be oriented? • Do I have to keep track of the solution if I need to backtrack to output the actual solution instead of the optimal result? • Should I use memoization? • Is there an important space constraint? Christopher Moh 2005

12. Implementation • Graph solutions • Should I use an adjacency list or a matrix? • Space and time constraints • Is my graph formulation solvable in polynomial time? • How many vertices/edges does my formulation produce? Is there a way to compress vertices/edges to improve runtime while still preserving accuracy of output? Christopher Moh 2005

13. Implementation • Brute Force solutions • What pruning methods are available? • Are there ways to generate output without going through redundancy or irrelevant search areas? [ See IOI 2001 Day 2 Q3 ] • Can a simple backtracking work? Or perhaps more advanced search methods e.g. Iterative Deepening, A* search are needed? Christopher Moh 2005

14. Testing • Testing for Accuracy with extreme cases • Isolated vertices? • Complete graphs? • Completely negative numbers? • Testing for Speed • How fast does the solution run when faced with a huge input? Christopher Moh 2005