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CSC 301.01, Class 37: Dynamic programming (3)

Overview

  • Preliminaries
    • Notes and news
    • Upcoming work
    • Extra credit
    • Questions
  • Review: Core ideas of dynamic programming
  • The edit distance problem
  • Formulating a solution
  • Varying the solution

News / Etc.

  • Warning! Our Chair is visiting class this week.
  • I hope you had a great Turkey break. I did not complete exam grading. (I think it’s currently third in my work queue.)

Upcoming work

  • Homework 9 due Wednesday.
  • Only one more homework after that one! (Dynamic programming.)
  • Final is in-class.

Extra Credit (Academic/Artistic)

  • CS Table Tuesday (unknown topic)
  • CS Extras Thursday (unknown topic)

Extra credit (Peer)

  • Swim meet this coming weekend.
  • Chamber Ensembles Saturday at 4:00 p.m.
  • Pub-free quiz

Extra Credit (Misc)

  • Forum: A Community Responds — A Bias/Hate Incident Raising Issues of Free Speech and Inclusivity. Tuesday, 11 a.m., Sebring-Lewis.
  • Mental Health Campus Resource Fair. Friday at 4pm in JRC 209.

Other good things

  • Jazz Ensemble Concert Friday at 7:30 p.m.
  • Collegium Concert Sunday at 2:00 p.m.

Questions

Review: Core ideas of dynamic programming

  • Typically used for optimization problems (sometimes involving exhaustive search)
  • Key idea: Build a table of intermediate results.
    • Need to design the table
    • Need to figure out the relationship of cells to other cells. (Recursive formulation of the solution.)
  • Example: For 0-1 knapsack, the great idea was a two dimensional table, with cell (i,j) gives the best way to make weight of i with elements 0…j. (approximately)
  • Generally gives us algorithms that run in the size of the table.

The edit distance problem

  • Input:
    • Two strings (source and target)
      • source has the form s[1] … s[n].
      • target has the form t[1] … t[m].
    • Three cost functions
      • Cost of deleting a character
      • Cost of inserting a character
      • Cost of replacing a character
  • Goal: Find a sequence of transformations that converts source to target with the minimum cost.
  • Note: Each cost function is typically constant, but we could permit them to vary based on the character we are inserting or deleting or even the position of the character.

An example

What’s the best way to transform abbababa to bbaababb

  • Given that insertion, deletion, and replacement all cost 1.
  • Given that deletion costs 1, insertion costs 2, and replacement costs 4.
    • Hint: Don’t replace.

Strategy for the first one?

  • Three replacements, cost of 3

Strategy for the second one

  • Delete the first a gives bbababa, cost of 1
  • Add an a after the second b, gives bbaababa, total cost of 3
  • Delete last a, cost of 1, gives bbaabab, total cost of 4
  • Add a b at the end, cost of 2, gives bbaababb, total cost of 6

Formulating a solution

Dynamic programming suggests that we should

  • Design a table. (What goes in each cell, what does it represent?)
  • Design a formula for filling in the cells of the table. (How do we compute a cell based on prior cells?)

So, let’s try.

  • Remember that our goal is to find the minimum edit distance from s to t. (And, optionally, the operations we need for that edit distance.)

C[i,j] stands for minimum cost of converting s[0..i] to t[0..j].

i is a column, j is a row

Goal 0: Define column 0 and row 0

Goal 1: Define C[i,j] in terms of C[i’,j’] where i’<i and/or j’<j

Note: Deletion costs d, Insertion costs a, replacement costs r

C[0,0] = 0
C[i,0] = d + C[i-1,0] (for i>0)
     * Permit C[i,0] = d(i,s[i]) + C[i-1,0]
     * You generally won't worry about cost functions, but it's useful
       to write formulate that accommodate them
C[0,j] = a + C[0,j-1] (for j>0)
C[i,j] = if (s[i] == t[j]) then
           min(d+C[i-1,j], C[i-1,j-1], a+C[i,j-1]) 
         else 
           min(d+C[i-1,j], r+C[i-1,j-1], a+C[i,j-1]) 

How long does this take to run? O(mn). Not great, but not bad.

Varying the problem

What if we just care about the best position of source within target?