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DP table

Build a DP table for matrix chain.

DP table

The two-dimensional table m[i][j] stores the lowest cost of multiplying matrices from i to j. The diagonal m[i][i] = 0 because a single matrix needs no multiplication.

The table solves subproblems and reuses results in larger problems. This is the core of dynamic programming.

Table size

For n matrices the table is n × n. Initially all values are zero; not just the diagonal, every cell is zeroed.

Then subproblems are solved to fill the table. Short chains are computed first, then longer chains.

Practice init_table function

Write init_table(n). Return an n×n zero matrix. Print it with print(init_table(4)).

Starter code

solution.py
def init_table(n):
    # Return n×n zero matrix

Sample solution

solution.py
def init_table(n):
    # Return n×n zero matrix
    return [[0] * n for _ in range(n)]
print(init_table(4))

Expected output

[[0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0]]

Questions and answers

Review the lesson questions together with their answers. To test yourself, solve the lesson interactively.

1 Diagonal value

Question 
m[i][i] = ____
Answer 
m[i][i] = 0

Explanation: 0 is right.

2 What is the table size for three matrices?

  • 3×3
  • 2×2 Correct answer
  • 4×4
  • 5×5

Explanation: Three matrices mean four dimensions, so the table is 4×4.

3 DP table: match the terms

Term Meaning
m[i][i] 0
m[1][2] 6000
m[2][3] 24000
m[1][3] 18000

All lessons in Advanced

1 DP - Memoization

  1. What is memoization?
  2. Fibonacci memoization
  3. Dict-based cache
  4. Summary - Memoization

2 DP - Tabulation

  1. What is tabulation?
  2. Building the table
  3. Fibonacci tabulation
  4. Summary - Tabulation

3 LCS

  1. LCS definition
  2. DP table
  3. Backtracking the table
  4. Summary - LCS

4 Knapsack

  1. Knapsack problem
  2. 0/1 knapsack DP
  3. Decision table
  4. Summary - Knapsack

5 DP project

  1. Project: Overview
  2. Cost table
  3. DP solution
  4. Print the result

6 Matrix chain

  1. Multiplication order
  2. DP table
  3. Split point
  4. Summary - Matrix chain

7 Floyd-Warshall

  1. All pairs
  2. Triple loop
  3. Table update
  4. Summary - Floyd-Warshall

8 Dijkstra

  1. Single source
  2. Priority queue
  3. Relaxation
  4. Summary - Dijkstra

9 Bellman-Ford

  1. Negative weights
  2. Edge relaxation loop
  3. Negative cycle
  4. Summary - Bellman-Ford

10 Kruskal

  1. Minimum spanning tree
  2. Edge sorting
  3. Cycle check
  4. Summary - Kruskal

11 Prim

  1. Prim idea
  2. Priority queue
  3. Visited set
  4. Summary - Prim

12 Topological sort

  1. Dependency order
  2. DFS-based sort
  3. Kahn algorithm
  4. Summary - Topological sort

13 Union-find, trie, segment tree

  1. Union-find
  2. Trie
  3. Segment tree
  4. Summary - Advanced structures

14 Advanced final

  1. Challenge 1
  2. Challenge 2
  3. Challenge 3
  4. Challenge 4