Divide and Conquer Algorithms Part III

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Divide and Conquer AlgorithmsPart III

• Vasileios Hatzivassiloglou
• University of Texas at Dallas

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Solving divide and conquer recurrences

• Theorem 1 Let f(n) be an increasing function
  with f(n) af(n/b) c whenever n is divisible
  by b, where a1, b integer greater than 1, and c
  a positive real. Then,
• if a1, f(n) O(logn)
• if agt1, f(n) O(nlogba)
• Additionally, the theorem specifies the exact
  values of f(n) for n a power of b.

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Proving Theorem 1

• If nbk for some k (k logbn),
• f(n) af(n/b) c
  a(af(n/b2) c) c
  a2f(n/b2) ac c
  a2(f(n/b3) c) ac c
  a3f(n/b3) a2c ac
  c ... akf(1) c(1 a a2
  ... ak-1)
• Assume a1 and that nbk. Then, ai 1 and the
  sum is ck, so that f(n) f(1) ck f(1)
  clogbn O(logn).

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Proving Theorem 1

• If a1 but n is not a power of b, there is some k
  such that bk lt n lt bk1. Because f is increasing,
  f(bk) f(n) f(bk1).
• f(n) f(bk1) f(1) c(k1) f(1) c ck lt
  f(1) c clogbn since k lt logbn. So f(n) is
  again O(logn).

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Proving Theorem 1

• Now consider agt1 and first the case where nbk
  for some k.
• The sum is c(ak-1)/(a-1) (sum of the terms of a
  geometric progression), so
• f(n) akf(1) c(ak-1)/(a-1)
  (f(1)c/(a-1))ak c/(a-1) O(ak)
  O(alogbn) O(nlogba).
• The proof for the remaining case is similar to
  the previous part of the proof, using the next
  higher k.

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Applying Theorem 1

• Binary search
• f(n)f(n/2)2
• O(logn)
• Finding the minimum and maximum
• f(n)2f(n/2)2
• O(nlog22) O(n1) O(n)

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Reading

• Section 7.3 (Theorem 1 to Example 8)

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Review problems

• Find the asymptotic form (O notation) of an
  increasing function f for which f(n) 3f(n/3)
  3 when n is divisible by 3.
• Solve the recurrence relation T(n) 3T(n/2) 2,
  T(1) 2 when n is a power of 2.
• An elimination tournament has n 2k teams in
  each round, all surviving teams play exactly one
  game and the winners advance to the next round.
  Construct recurrence relations for (a) the number
  of rounds played (b) the total number of games
  played.