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CMSC 203 / 0201 Fall 2002

CMSC 203 / 0201 Fall 2002. Week #10 – 28/30 October and 1 November 2002 Prof. Marie desJardins. TOPICS. Recurrence relations and solutions Divide-and-conquer recurrences. MON 10/28 RECURRENCE RELATIONS (5.1). Concepts/Vocabulary. Recurrence relations Solution / solution sequence

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CMSC 203 / 0201 Fall 2002

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  1. CMSC 203 / 0201Fall 2002 Week #10 – 28/30 October and 1 November 2002 Prof. Marie desJardins

  2. TOPICS • Recurrence relations and solutions • Divide-and-conquer recurrences

  3. MON 10/28 RECURRENCE RELATIONS (5.1)

  4. Concepts/Vocabulary • Recurrence relations • Solution / solution sequence • Initial conditions • Useful examples: compound interest, bunny rabbits / Fibonacci, Tower of Hanoi, Catalan numbers

  5. Examples • Exercise 5.1.4: For each of the following sequences, find a recurrence relation satisfied by this sequence: • (b) an = 2n • (d) an = 5n • (f) an = n2 + n • (g) an = n + (-1)n

  6. Examples II • Save early and often: Exercise 5.1.6: A person deposits $1000 in an account that yields 9% interest compounded yearly. • (a) Set up a recurrence relation for the amount in the acount at the end of n years. • (b) Find an explicit formula for the amount in the account at the end of n years. • (c) How much money will the account contain after 100 years?

  7. Examples III • Exercise 5.1.11: Use mathematical induction to verify the formula derived in Example 5 for the number of moves required to complete the Tower of Hanoi puzzle: • Hn = 2n - 1 • Catalan numbers: Example 5.1.8 (p. 315): Find a recurrence relation for Cn, the number of ways to parenthesize the product of n+1 numbers, x0x1…xn. • Cn = k=0n-1 Ck Cn-k-1 • C0 = C1 = 1

  8. Examples IV • Exercise 5.23: A ternary string contains only 0s, 1s, and 2s. • (a) Find a recurrence relation for the number of ternary strings that do not contain two consecutive 0s. • (b) What are the initial conditions? • (c) How many ternary strings of length six do not contain two consecutive 0s? • Exercise 5.25: • (a) Find a recurrence relation for the number of ternary strings that do not contain two consecutive 0s or two consecutive 1s. • (b) What are the initial conditions? • (c) How many ternary strings of length six do not contain two consecutive 0s or two consecutive ones?

  9. WED 10/30 - FRI 11/1DIVIDE-AND-CONQUER (5.3) ** HOMEWORK #7 DUE ** ** FEEDBACK SESSION TODAY **

  10. Concepts / Vocabulary • Divide-and-conquer recurrence relations • f(n) = a f(n/b) + g(n)

  11. Examples • Exercise 5.3.7: Suppose that f(n) = f(n/3) + 1 when n is divisible by 3, and f(1) = 1. Find • (a) f(3) • (b) f(27) • (c) f(729) = f(36) • Fast multiplication (Example 5.3.3, p. 333) • ab = (22n + 2n) A1B1 + 2n(A1-A0)(B0-B1) + (2n+1)A0B0 • Exercise 5.3.3: Multiply (1110)2 and (1010)2 using the fast multiplication algorithm. • Exercise 5.3.4: Express the fast multiplication algorithm in pseudocode.

  12. Examples II • Exercises 5.3.14-16: • Suppose that there are n = 2k teams in an elimination tournament, where there are n/2 games in the first round, with the n/2 = 2k-1 winners playing in the second round, and so on. Develop a recurrence relation for the number of rounds in the tournament. • How many rounds are there when there are 32 teams? • Solve the recurrence relation for the number of rounds in the tournament.

  13. Examples III • Theorem 1: for a recurrence relation f(n) = a f(n/b) + cwhenever b|n, a1, integer b>1, and real c>0, f(n) = O(nlogba) if a > 1 and f(n) = O(log n) if a=1.

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