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MIDTERM ECE750 CRIBSHEET 1. B ASIC NOTATIONS x x {x} x - x log x log2 x ln x loge x n greatest integer x ("floor") least integer x ("ceiling") "decimal part" n! = k=1 k a! (a - b)! b! factorial a b = 2. A SYMPTOTICS f g means limn f f g means g means f (n) g(n) =1 for some positive c1 , c2 lim supn f (n) < g(n) f (n) [c1 , c2 ] eventually, g(n) f O(g) f f o(g) f f (g) f f (g) f f (g) f Useful rules: g g g g g (1) O(g(n))O(f (n)) = O(f (n)g(n)) (2) If f g, then O(f (n)) + O(g(n)) = O(g(n)). For a C function f , f (x) = f (0) + xf (0) + x2 x3 f (0) + f (0) + 2! 3! 1 2 ECE750 MIDTERM CRIBSHEET Common asymptotic expansions: b b2 - 2 + a a b ln a + a b b2 log2 (a + b) = log2 a + - 2 + a ln 2 a ln 2 b log2 a + a 1 1 log a! a log a - ln 2 a + 2 log a + log 2 + (a-1 ) ln(a + b) = ln a + x x + (1) e log x+ if b a if b a ex + O(ex ) b a if o(1) 1 + log a - log b - O(log b) when b ln 2 n3 m! n2 n ln n ln m - + -O (m - n)! 2m 2m m2 3. T IME ANALYSIS AND a b Z-TRANSFORMS t1 for i = 1 to n . end t2 (i) . . if t1 t2 n t1 = c1 + i=1 (c2 + t2 (i)) t3 e then . . . t1 = t2 + c1 + max(t3 , t4 ) . . . else t4 (1) (2) Z Z[f (n)] = n=0 -1 f (n)z -n 1 2i [F (z)] = F (z)z n-1 dz C Z [T (n - a)] = z -a T (z) shift Z [(n)] = 1 impulse 1 Z [u(n)] = 1-z-1 step function 1 Z [an ] = 1-az-1 z -1 Z [n] = (1-z-1 )2 Z n2 = z -1 +z -2 (1-z -1 )3 ECE750 MIDTERM CRIBSHEET 3 The delta and step functions (note: functions are N R): (n) = 1 if n = 0 0 otherwise 1 if n 0 0 otherwise u(n) = Theorem 3.1 (`Master'). The solution to a divide-and-conquer recurrence the of form T (n) = kT ( n/s ) + (nd ) where s > 1, is n log k log s T (n) = nd log n (nd ) 4. D ATA if sd < k if sd = k if sd > k STRUCTURES Fields of common nodes in data structures: Linked list: data, next Bidirectional linked list: data, next, prev Binary tree: data, lchild, rchild Red-black tree: data, lchild, rchild, colour Heap: priority, lchild, rchild Treap: data, lchild, rchild, priority Trie: lchild, rchild; leaves contain a data field. Catalan numbers. The number of binary trees with n nodes is Cn = 2n 1 1 4n 3/2 n n+1 n Theorem 4.1. The expected height of a binary search tree constructed by inserting a sequence of n random values is c log n with c 4.311. DA A A A E C rotate left A ~ ~ ~ ~~ BA A A A } } } ~} rotate right A ~ ~ ~ ~~ BA A A A } } } ~} DA A A A E C Red-black trees. In a red-black tree, bh(x) is the number of black nodes along any path from a node x to a leaf, excluding the leaf. The number of internal nodes in the subtree rooted at x is at least 2bh(x) - 1. A red-black tree with n internal nodes has height at most 2 log2 (n + 1). Theorem 4.2 (Kraft inequality). There exists a uniquely decodeable code (equivalently, a trie) with codeword lengths D = {d1 , d2 , } (equivalently, path lengths in a trie from root to leaves), if and only if 2-d 1 dD 4 ECE750 MIDTERM CRIBSHEET F IGURE 1. Rebalancing of red-black trees. with equality when every internal node of the trie has two children. Hash functions. Let m be the size of the hash table, and k a key. Two common hash functions are: h(k) = x mod m h(k) = m{k} with irrational

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