CS 124: Course Review for Final Exam. Grades So Far (with Top Students) Exam Format: Similar to Midterms Class Quizzes Some Topic Highlights

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1 S : ourse Review for inal xam Grades So ar (with Top Students) xam ormat: Similar to Midterms lass Quizzes Some Topic Highlights Illustrations of O,, (lass Quiz ) f (N) g (N) g (N) True or false?. g (N)=O(f (N)). g (N)=O(f (N)). g (N)=O(f (N)). g (N)= (f (N)) g (N) 5. g (N)= (f (N)) n 0 f (N) N. g (N)= (f (N)) True in all cases. lass Quiz Outcome 00%: (out of 5)

2 rithmetic expression tree lass Quiz : Traverse this tree into (a) prefix, (b) infix, and (c) postfix expressions. + * / + - * lass Quiz Outcome 00%: (out of 5) What if the access path length is an odd number? One rotation must be a single rotation. o it either at the bottom or at the top. The resulting tree structures may be different. oth are fine! lass Quiz Outcome 00%: (out of 5) lass Quiz : Let s do it at the bottom in this class (with T,S,,M,O,N).

3 ouble hashing: example fter inserting the first 0 items k =,,, 0,,,, 5,,. Inserting the next items: k = 5,, 5, using h i (k) = (h(k) + i (k % + ) % 0 lass Quiz : lass Quiz Outcome 00%: (out of ) 5 Shellsort: lass Quiz 5 Sort the file ILOVRURSION using the increment sequence,. Total #comps: Total #swaps: lass Quiz 5 Outcome 00%: (out of ) Phase (h=): LINOIORSVUR (;) Phase (h=): IILNOORRSUV (;) nswers: ;

4 Quicksort: lass Quiz Sort the file ILOVRURSION. hoose the median-of-three as the pivot (and show it in each recursion). I L O V R U R S I O N pivot: {I,N,} -> I I I R O L U R S N O V I I R O L U R S N O V I pivot: {I,,} -> R O L U R S N O V pivot: {R, V, R} -> R I I L N O O R R S U V lass Quiz Outcome 00%: (out of ) lass Quiz : S and S readth-first search and depth-first search starting with vertex : List the orders of vertices to be visited on (a). (Hint: when there is more than one adjacent node to choose, pick the smallest numbered node.) lass Quiz Outcome 00%: (out of 5) S: {,,,,5,,,} S: {,,,5,,,,} (a) (b)

5 S : ourse Review for inal xam Grades So ar (with Top Students) xam ormat: Similar to Midterms lass Quizzes Some Topic Highlights xercise: (hapter, Slide ) 0 ase case (n=): Verify LS = RHS Inductive hypothesis (n=k-): ssume LS = RHS When n=k, Prove LS = RHS 5

6 xample: binary search (hapter ) // Search for x in an array a[i..j] sorted in the ascending order. // If found then return the array index m such that a[m] = x // else return NOT_OUN; int a[]; binsearch(x, i, j) { m = floor((i+j)/); if (x == a[m]) then return m; else if (i == j) then return NOT_OUN; else if (x < a[m]) then return binsearch(x, i, m-); else return binsearch(x, m+, j); } Run-time: the number of comparisons between array elements Problem size: the number of elements in the array Recurrence equation: T(N) = T(N/) + if N >= = if N =. Let N = n. Then, recursively, T( n ) = T( n- ) + T( n- ) = T( n- ) +.. T( ) = T() + T() = T() ( + T( n ) = T() + n = + n Hence, T(N) = + logn = O(logN)

7 - arrays vs. - multi-lists for a matrix (hapter ) Let m, n, and k be the numbers of columns, rows, and non-zero elements, respectively. Then: - array Multi-list Storage space (mn) (m+n+k) Row access time (m) O(m) olumn access time (n) O(n) lement access time () O(k) multi-list is better if the matrix is sparse. Summary: rotations in VL and splay (hapter ) Tree type ase zig-zig (left-left or right-right) zig-zag (otherwise) VL tree single rotation bottom-up double rotation Splay tree top-down double rotation (the roots) bottom-up double rotation (the current node)

8 omparison of collision resolution schemes (hapter 5) Separate chaining Use of links - costs memory and calls on memory allocation Hence typically is expensive. Linear probing asily implementable Performance degrades severely as the load factor increases, due to primary clustering. Quadratic probing Slightly more difficult to implement with good performance n insertion can (but not likely) fail if the table is half empty. ouble hashing liminates primary and secondary clustering; The computation of a second hash function can be costly. 5 Sorting: Summary (hapter ). Selection sort. ubble sort. Insertion sort (Section.). Shellsort (Section.) 5. Heapsort (Section.5). Mergesort (Section.). Quicksort (Section.). ounting sort. Radix sort (Section.) O(N ) O(N logn) 0. xternal sorting (Section.) O(N) one of the most creative S topics!

9 Graph traversal: algorithms (hapter ) readth-first search (or traversal) epth-first search (or traversal) (We consider connected graphs here.) ijkstra s algorithm: example (/) Start vertex: ck: lifford Shaffer for the graph chosen candidate discarded

10 0 ijkstra s algorithm: example (/)