COSC 2006 Data Structures I Recursion III

kth smallest Can be solved by sorting, but this does more work than needed (why?) solving kth smallest for all k More efficient solution possible: Pick an element, any element partition the other elements into two sets: those smaller (S1) and those bigger (S2) If S1 has k-1 members, we got lucky! If S1 has at least k members, recurse to find the kth smallest in S1 Otherwise recurse to find the kth smallest in S2 …

kth smallest: Devil in the details What about this last case? We dispensed with |S1| + 1 elements! So we are looking for the ___th smallest element in S2 (k - (|S1| + 1))th Need to “pass” subarray with first, last So, |S1| = pivotIndex - first

kth smallest: partial code This motivates the details: kSmall(k, anArray, first, last) choose a pivotIndex if (k < pivotIndex - first + 1) return kSmall(k, anArray, first, pivotIndex-1) else if (k == pivotIndex - first + 1) return p else return kSmall(k - (pivotIndex-first+1), anArray, pivotIndex+1, last)

Problems with Recursion Memory (stack) exhaustion: if many stack frames are pushed - can run out of space for stack Time: each recursive call requires time to set up a new stack frame, copy in actual parameter values and transfer execution

Problems with Recursion Any recursive algorithm can be rewritten using iterative technique - it will be longer and more likely to have problems but will run faster.

Recursion Vs Iteration Note that just because we can use recursion to solve a problem, that does not mean this is the optimal solution. For instance, we usually would not use recursion to solve the sum of 1 to N problem, because the iterative version is easier to understand and requires less memory is N is a large number.

Recursion Vs Iteration The iterative solution is easier to understand: sum = 0; for(int number = 1; number <=N; number++) sum += number; Recursion has memory overhead. However, for some problems, recursion provides an elegant solution, often cleaner than an iterative version

Example: Towers of Hanoi For some problems, recursive solution is the only reasonable solution Given: N disks & three poles: A the source B the destination C the spare (temporary) The disks are of different sizes & have holes in the middle Only one disk can be removed at a time A larger disk can't be placed on a smaller one Initially all disks are on pole A Required: Move the disks, one-by-one, from pole(peg) A to B, using pole C as spare A B C

Example: Towers of Hanoi Solution: Only one disk (N=1, First base case): Move the disk from the source peg A to the destination peg B Only two disks (N=2, Second base case): Move the top disk from peg A to peg C Move the second disk from peg A to peg B Move the top disk from peg C to peg B

Example: Towers of Hanoi Solution: Three disks (N=3): Move top disk from peg A to peg B Move second disk from peg A to peg C Move top disk from peg B to peg C Move the remaining bottom disk from peg A to peg B Move the top disk from peg C to peg A Move second disk from peg C to peg B A C B

Towers of Hanoi 1- initial stage 2- move 1 disk from A to C using B 3- move 1 disk from A to B using C A C B A C B A C B 1 2 3

Towers of Hanoi - Stages C B A C B A C B 4 5 6 4- move 1 disk from C to B using A now have moved 2 disks from A to B using C 5- move 1 disk from A to C using B 6- move 1 disk from B to A using C 7- move 1 disk from B to C using A 8- move 1 disk from A to C using B completes moves of 2 disks from B to C using A and completes moves of 3 disks from A to C using B A C B A C B 7 8

Example: Towers of Hanoi Solution: General Solution (N-1) disks: Move the top (N-1) disks to Temp pole Move the bottom disk to the destination Move the N-1 disks from Temp to the destination

Example: Towers of Hanoi Solution: General Solution (N-1) disks: a) Initial state b) Move N-1 disks from A to C c) Move 1 disk from A to B d) Move N-1 disks from C to B A B C

Towers of Hanoi First let’s play with blocks … solveTowers(count, source, dest, spare) if (count is 1) { move a disk directly from source to dest else solveTowers(count-1, source, spare, dest) solveTowers(1, source, dest, spare) solveTowers(count-1, spare, dest, source)

Towers of Hanoi - Algorithm public static void solveTowers(int n, char source, char dest, char intermed) { if (n == 1) System.out.println("move a disk from " + source + " to " + dest); else solveTowers(n-1, source, intermed, dest); solveTowers(1, source, dest, intermed); solveTowers(n-1, intermed, dest, source); } original call: solveTowers ( 3, 'A', 'C', 'B'); 2 Trace: n = 3 S = A D = C I = B ret = 1 n = 2 n = 2 S = A S = B D = B D = C I = C I = A ret = 2 ret = 3 n = 1 n = 1 n = 1 n = 1 S = A S = C S = B S = A D = C D = B D = A D = C I = B I = A I = C I = B ret = 2 ret =3 ret =2 ret = 3 3 1

Example: Towers of Hanoi Observations Solution analog to mathematical induction If we can solve the problem for N-1 disks, then we can solve it for N disks In each move, the source, destination, and temporary poles change locations Three recursive calls inside the function

Review When you solve a problem by solving two or more smaller problems, each of the smaller problems must be ______ the base case than the original problem. closer to farther to either closer to or the same “distance” from either farther to or the same “distance” from

Review In a sorted array, the kth smallest item is given by ______. anArray[k-1] anArray[k] anArray[SIZE-k] anArray[SIZE+k]

Review A recursive solution that finds the factorial of n always reduces the problem size by ______ at each recursive call. 1 2 half one-third

Review In the recursive solution to the kth smallest item problem, the problem size decreases by ______ at each recursive call. 1 at least 1 half at least half

Review In the recursive solution to the Towers of Hanoi problem, the number of disks to move ______ at each recursive call. decreases by 1 increases by 1 decreases by half increases by half

Review A recursive solution that finds the factorial of n generates ______ recursive calls. n-1 n n+1 n*2

Review In the Fibonacci sequence, which of the following integers comes after the sequence 1, 1, 2, 3? 3 4 5 6