Assignment 5 Debrief

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Assignment 5 Debrief

• Andy Wang
• Data Structures, Algorithms, and Generic
  Programming

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Word Ladder Game

• Idea Find a way to transform one word to
  another through words that are one character away
• Example bunny ? tiger
• bunny ? funny ? funky ? funks ? finks ? fines ?
  tines ? tiles ? tiler ? tiger

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Bunny to Tiger?
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Brute Force Approach
start

• n number of words in a dictionary
• Speed complexity O(ad), a 526
• Space complexity O(n)

atart..ztart, saart..smart..szart, start..stzrt,
staat..stazt, stara..starz
d
amart..zmart, saart..szart, smart..smzrt,
smaat..smazt, smara..smarz

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Brute Force Approach

• Speed complexity (265)d
• Suppose d 4, we need 300,000,000 steps
• Space complexity
• 5 char/word 5000 words 25,000 characters

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Adjacency Graph Approach

• Idea go through only words that are one
  character away

dear
bear, fear,.., year, dear, dead
dear, fear, year, boar, beer, beadbeap

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Adjacency Graph Approach

• Need to build an adjacency graph
• bear dear, fear, year, boar, beer, beadbeap
• dear bear, fear,.., year, dear, dead
• Need to avoid revisiting the same words
• Do a BFS to find the shortest path

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Building the Adjacency Graph

• For each word, do pair-wise comparisons against
  all words
• If a word is one character away, append to its
  list
• n number of words in a dictionary
• Speed complexity O(n2)
• Space complexity O(n2)

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Building the Adjacency Graph

• Speed complexity
• 5 comparisons to determine a word is one
  character away
• For each word, it needs to perform 5
  comparisons/word 5,000 words 25,000
  comparisons
• For 5,000 words, we need 125,000,000 comparisons
• best case average case worst case

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Building the Adjacency Graph

• Space complexity
• If every word is one character away from every
  word in a dictionary
• We need 5 char/word 5,000 words 5,000 words
  125,000,000 characters (worst case)
• Average case 130,000 characters (from
  empirical measurements)

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Can We Do Better?

• O(n) speed?

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Visualizing the Solution Space

• Try a simpler case
• Three-letter words
• Visualize cat

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Visualizing (c, a, t)
a
c
(c, a, t)
t
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Visualizing (c, a, t)
a
c
r
(c, a, t)
(r, a, t)
t
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Visualizing (c, a, t)

• You can make words that are one character apart
  by collapsing one of the dimensions

(b, a, t)
(r, a, t)
(c, a, t)
a
t
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Collapsing the Solution Space?

• Idea Use hashing
• Create mapltstring, setltstringgtgt
• For cat, hash the following
• at
• ct
• ca
• After processing bat, cat, and rat
• mapat will contain bat, cat, and rat

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Hash-Based Graph Construction

• Speed complexity O(n)
• 5 char/word 5000 words 25,000 hashing
  operations
• Space complexity O(n)
• 5 char/word 5000 words 25,000 characters

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Modified BFS

• Each word expands into several lists

dear
ear, dar, der, dea
(bear, fear,.., year, dear, dead)
ear, bar, ber, bea (dear, fear, year, boar,
beer, beadbeap)

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BFS Complexity

• Speed complexity O(n edges)
• Since each word is visited at most once
• Storage complexity O(n)