1. A Dynamic Programing Based Solution to the Two-Dimensional Jump-It Problem
Jamil Saquer and Razib Iqbal
Computer Science Department
Missouri State University
Springfield, MO

2. Introducing the problem
2D board with a cost to visit each cell
A player starts at the top-left cell and wants to reach the bottom-right (exit) cell with the lowest cost
Allowed moves:
one cell to the right
one cell below
jump over one cell to the right
Jump over one cell below 5 7 2 4 8 1 6 3 9

3. Possible Paths 5 7 2 4 8 1 6 3 9 5 7 2 4 8 1 6 3 9
cost = = 20
cost = = 29 5 7 2 4 8 1 6 3 9
cost = = 18

4. Goal Find the cheapest cost of playing the game
Find path that leads to playing the game with the cheapest cost

5. Review – 1D Jump-It 5 15 75 7 43 11

6. Review – 1D Jump-It solution: if board length == 1, visit that cell 5
start
solution:
if board length == 1, visit that cell 5 15 75 7 43 11

7. Review – 1D Jump-It solution: if board length == 1, visit that cell
start
solution:
if board length == 1, visit that cell
if board length == 2, visit both cells 5 15 75 7 43 11

8. Review – 1D Jump-It solution: if board length == 1, visit that cell
start
solution:
if board length == 1, visit that cell
if board length == 2, visit both cells
if board length == 3, cheaper to jump over 5 15 75 7 43 11

9. Review – 1D Jump-It solution: if board length == 1, visit that cell
start
solution:
if board length == 1, visit that cell
if board length == 2, visit both cells
if board length == 3, cheaper to jump over
else min_cost = board[i] min{jumpIt(i+1), jumpIt(i+2)} 5 15 75 7 43 11

10. 2D Jump-It Finding the Cheapest Cost
Let jumpIt(r, c) be the cheapest cost of playing the game starting at any cell (r, c)
Many cases 5 7 2 4 8 1 6 3 9

11. Finding the Cheapest Cost5 7 2 4 8 1 6 3 9

12. Finding the Cheapest Cost5 7 2 4 8 1 6 3 9

13. Dynamic Programming Solution
2D jump-It is a good candidate for DP
Overlapping sub-problems
Optimal sub-structure

14. Top-Down DP Solution board[i][j] contains cost of visiting cell (i, j)
costs – a cache table for storing solutions
costs[i][j] minimum cost starting game at cell (i, j)
fill cache table starting with basic cases 5 7 2 4 8 1 6 3 9
costs
board

15. Top-Down DP Solution costs board fill last row starting
fill in last column
fill the row before last
fill the column before last
fill rest of cache table 18 18 13 11 14 5 7 2 4 8 1 6 3 9 14 13 14 7 6 21 12 16 16 11 14 11 10 12 3
costs
board

16. Finding Optimal path board costs path
Use another cache table, path, to remember moves
path[i][j] - coordinates of cell visited after cell (i ,j) 1 2 3 4 5 7 2 4 8 1 6 3 9 18 18 13 11 14 14 13 14 7 6 1 21 12 16 16 11 2 14 11 10 12 3 3
board
costs
(0, 2)
(0, 3)
(0, 3)
(1, 3)
(1, 4) 1
(1,1)
(1, 3)
(1, 4)
(1, 4)
(3, 4) 2
(2, 1)
(3, 1)
(3, 2)
(2, 4)
(3, 4) 3
(3, 2)
(3, 2)
(3,4)
(3, 4)
(-1,-1) 1 2 3 4
path

17. 2D Jump-It in The CS Curriculum
Good problem to use when teaching DP

18. Questions