CSE 373: Data Structures and Algorithms. Binary Search Trees. Autumn Shrirang (Shri) Mare

Transcription

1 CSE 373: Data Structures and Algorithms Binary Search Trees Autumn 2018 Shrirang (Shri) Mare Thanks to Kasey Champion, Ben Jones, Adam Blank, Michael Lee, Evan McCarty, Robbie Weber, Whitaker Brand, Zora Fung, Stuart Reges, Justin Hsia, Ruth Anderson, and many others for sample slides and materials...

2 Administrivia Due dates: - HW2 Part 1 due today 11:59pm - If you are submitting late, fill the late submission form (link available on the website) - Double check that you are using the right checkstyle (CSE373-checkstyle) - See Step 2b in Eclipse setup ( - See HW2 Part 0b Notes - Any questions on HW2 Part 1 logistics? - You can continue working on HW2 Part 2 (i.e., pushing commits to your repository). It won t affect your HW2 Part 1 submission. - Section 2 solutions will be posted today 5pm (generally, we ll post Section solutions on Fridays around 5pm) CSE 373 AU 18 SHRI MARE 2

3 Modeling complex functions CSE 373 AU 18 SHRI MARE 3

4 <latexit sha1_base64="htkyuwrcgzq19tsodjf2cb58kn0=">aaab8hicbzdlssnafizpvnz4q7p0m1gevyxpxm6eghuxfexfklam00k7dgyszizccx0kny4ucevjupntnlzzaospax//oyc5548zzrtxvg9ny3nre2e3sufuhxwehvdptrs6zrwhhzlyvpvjrclnknymm5z2m0wxidntxzpbeb33rjvmqxww04xgao8ksxjbxlqpyedeiomg0aba8+reqmgd/bjquko9qh6fw5tkgkpdony68l3mravwhhfoz26ya5phmsejgliuwfadfyufz+jsokoupmo+addc/t1rykh1vms2u2az1qu1uflflchn0owkjrpcuemwhyu5ryzf8+vrkclkdj9aweqxuysiy6wwmtyj14bgr568dt1g3bd836i1mmucftihc7gch66hbxfqhg4qepamr/dmkoffexc+lq0btjlzbn/kfp4a3q2pfw==</latexit> <latexit sha1_base64="htkyuwrcgzq19tsodjf2cb58kn0=">aaab8hicbzdlssnafizpvnz4q7p0m1gevyxpxm6eghuxfexfklam00k7dgyszizccx0kny4ucevjupntnlzzaospax//oyc5548zzrtxvg9ny3nre2e3sufuhxwehvdptrs6zrwhhzlyvpvjrclnknymm5z2m0wxidntxzpbeb33rjvmqxww04xgao8ksxjbxlqpyedeiomg0aba8+reqmgd/bjquko9qh6fw5tkgkpdony68l3mravwhhfoz26ya5phmsejgliuwfadfyufz+jsokoupmo+addc/t1rykh1vms2u2az1qu1uflflchn0owkjrpcuemwhyu5ryzf8+vrkclkdj9aweqxuysiy6wwmtyj14bgr568dt1g3bd836i1mmucftihc7gch66hbxfqhg4qepamr/dmkoffexc+lq0btjlzbn/kfp4a3q2pfw==</latexit> <latexit sha1_base64="htkyuwrcgzq19tsodjf2cb58kn0=">aaab8hicbzdlssnafizpvnz4q7p0m1gevyxpxm6eghuxfexfklam00k7dgyszizccx0kny4ucevjupntnlzzaospax//oyc5548zzrtxvg9ny3nre2e3sufuhxwehvdptrs6zrwhhzlyvpvjrclnknymm5z2m0wxidntxzpbeb33rjvmqxww04xgao8ksxjbxlqpyedeiomg0aba8+reqmgd/bjquko9qh6fw5tkgkpdony68l3mravwhhfoz26ya5phmsejgliuwfadfyufz+jsokoupmo+addc/t1rykh1vms2u2az1qu1uflflchn0owkjrpcuemwhyu5ryzf8+vrkclkdj9aweqxuysiy6wwmtyj14bgr568dt1g3bd836i1mmucftihc7gch66hbxfqhg4qepamr/dmkoffexc+lq0btjlzbn/kfp4a3q2pfw==</latexit> <latexit sha1_base64="htkyuwrcgzq19tsodjf2cb58kn0=">aaab8hicbzdlssnafizpvnz4q7p0m1gevyxpxm6eghuxfexfklam00k7dgyszizccx0kny4ucevjupntnlzzaospax//oyc5548zzrtxvg9ny3nre2e3sufuhxwehvdptrs6zrwhhzlyvpvjrclnknymm5z2m0wxidntxzpbeb33rjvmqxww04xgao8ksxjbxlqpyedeiomg0aba8+reqmgd/bjquko9qh6fw5tkgkpdony68l3mravwhhfoz26ya5phmsejgliuwfadfyufz+jsokoupmo+addc/t1rykh1vms2u2az1qu1uflflchn0owkjrpcuemwhyu5ryzf8+vrkclkdj9aweqxuysiy6wwmtyj14bgr568dt1g3bd836i1mmucftihc7gch66hbxfqhg4qepamr/dmkoffexc+lq0btjlzbn/kfp4a3q2pfw==</latexit> <latexit sha1_base64="3w/qnn1dbhnnim5hgdyj9goguha=">aaab8xicbzdlssnafibpek3xvnxpzrairkrsjd0ibtcuk9gljqfmppn26gqszizccx0lny4ucevbupntnlzzaospax//oyc5548ywbvx3w+0sbm1vbnb2xp2dw6pjqsnp12d5oqydk1fqvor0uxwytqgg8h6mwikiqtrrzpbeb33xjtmqxww04yfcrljhnnkjluea9+5wvq6qtio1ty6uxbeb6+egprqd6pfwtcleckkoyjo7xtuzskckmopydmnydxlcj2qefmtspiwhraljwf40jpdhkfkpmnwwv09uzbe62ks2c6emlferc3n/2p+bujmwhcz5yzjuvwozgu2kz6fj4dcmwre1akhittdmr0traixitk2bg/15hxonuqe5ftgrdus46jaovzafxhwds24gzz0gikez3ifn6trc3phh8vwdvtonmefoc8fmsapha==</latexit> <latexit sha1_base64="3w/qnn1dbhnnim5hgdyj9goguha=">aaab8xicbzdlssnafibpek3xvnxpzrairkrsjd0ibtcuk9gljqfmppn26gqszizccx0lny4ucevbupntnlzzaospax//oyc5548ywbvx3w+0sbm1vbnb2xp2dw6pjqsnp12d5oqydk1fqvor0uxwytqgg8h6mwikiqtrrzpbeb33xjtmqxww04yfcrljhnnkjluea9+5wvq6qtio1ty6uxbeb6+egprqd6pfwtcleckkoyjo7xtuzskckmopydmnydxlcj2qefmtspiwhraljwf40jpdhkfkpmnwwv09uzbe62ks2c6emlferc3n/2p+bujmwhcz5yzjuvwozgu2kz6fj4dcmwre1akhittdmr0traixitk2bg/15hxonuqe5ftgrdus46jaovzafxhwds24gzz0gikez3ifn6trc3phh8vwdvtonmefoc8fmsapha==</latexit> <latexit sha1_base64="3w/qnn1dbhnnim5hgdyj9goguha=">aaab8xicbzdlssnafibpek3xvnxpzrairkrsjd0ibtcuk9gljqfmppn26gqszizccx0lny4ucevbupntnlzzaospax//oyc5548ywbvx3w+0sbm1vbnb2xp2dw6pjqsnp12d5oqydk1fqvor0uxwytqgg8h6mwikiqtrrzpbeb33xjtmqxww04yfcrljhnnkjluea9+5wvq6qtio1ty6uxbeb6+egprqd6pfwtcleckkoyjo7xtuzskckmopydmnydxlcj2qefmtspiwhraljwf40jpdhkfkpmnwwv09uzbe62ks2c6emlferc3n/2p+bujmwhcz5yzjuvwozgu2kz6fj4dcmwre1akhittdmr0traixitk2bg/15hxonuqe5ftgrdus46jaovzafxhwds24gzz0gikez3ifn6trc3phh8vwdvtonmefoc8fmsapha==</latexit> <latexit sha1_base64="3w/qnn1dbhnnim5hgdyj9goguha=">aaab8xicbzdlssnafibpek3xvnxpzrairkrsjd0ibtcuk9gljqfmppn26gqszizccx0lny4ucevbupntnlzzaospax//oyc5548ywbvx3w+0sbm1vbnb2xp2dw6pjqsnp12d5oqydk1fqvor0uxwytqgg8h6mwikiqtrrzpbeb33xjtmqxww04yfcrljhnnkjluea9+5wvq6qtio1ty6uxbeb6+egprqd6pfwtcleckkoyjo7xtuzskckmopydmnydxlcj2qefmtspiwhraljwf40jpdhkfkpmnwwv09uzbe62ks2c6emlferc3n/2p+bujmwhcz5yzjuvwozgu2kz6fj4dcmwre1akhittdmr0traixitk2bg/15hxonuqe5ftgrdus46jaovzafxhwds24gzz0gikez3ifn6trc3phh8vwdvtonmefoc8fmsapha==</latexit> Some summations identities If system.out.println takes constant time 1, T(n) = #$%! 1 &'( = n If system.out.println takes constant time c, T(n) = #$%! c &'( = cn CSE 373 AU 18 SHRI MARE 4

5 Some summations identities T(n) = // n times T(n) = #$%! 1 = n &'( T(n) = c + c + c + c + // n times T(n) = #$%! c &'( #$% = c! 1 &'( = cn Summation of a constant T(n) = (n -1 ) T(n) = #$%! i &'( n n 1 = Gauss's identity 2 CSE 373 AU 18 SHRI MARE 5

6 Modeling complex loops: Simplifying summations +c i-1 n (0c + 1c + 2c + 3c + + i-1c) + (0c + 1c + 2c + 3c + + i-1c) + (0c + 1c + 2c + 3c + + i-1c) + repeat n times T(n) = #$% &$%!! c #$% =! ci &'( Summation of a constant #$% = c! i Factoring out a constant &'( = c n n 1 2 Gauss s Identity = c 2 na c 2 n O(n2 ) CSE 373 WI 18 MICHAEL LEE 6

7 Modeling recursion CSE 373 SP 18 - KASEY CHAMPION 7

8 Modeling recursion: Unfolding Method CSE 373 AU 18 SHRI MARE 8

9 Binary Search Trees CSE 373 AU 18 SHRI MARE 9

10 Storing Sorted Items in an Array get() O(logn) put() O(n) remove() O(n) Can we do better with insertions and removals? CSE 373 SP 18 - KASEY CHAMPION 10

11 Trees! A tree is a collection of nodes - Each node has at most 1 parent and 0 or more children 1 Root node: the single node with no parent, top of the tree Branch node: a node with one or more children Leaf node: a node with no children Edge: a pointer from one node to another Subtree: a node and all it descendants Height: the number of edges contained in the longest path from root node to some leaf node CSE 373 SP 18 - KASEY CHAMPION 11

12 Tree Height What is the height of the following trees? overallroot overallroot overallroot 1 7 null Height = 2 Height = 0 Height = -1 or NA CSE 373 SP 18 - KASEY CHAMPION 12

13 Traversals traversal: An examination of the elements of a tree. A pattern used in many tree algorithms and methods Common orderings for traversals: pre-order: process root node, then its left/right subtrees overallroot in-order: process left subtree, then root node, then right post-order: process left/right subtrees, then root node CSE 373 SP 17 ZORA FUNG 13

14 Binary Search Trees A binary search tree is a binary tree that contains comparable items such that for every node, all children to the left contain smaller data and all children to the right contain larger data CSE 373 SP 18 - KASEY CHAMPION 14

15 Binary Search Trees 10 foo Binary Search Trees allow us to: - quickly find what we re looking for - add and remove values easily 7 bar 12 baz Dictionary Operations: Runtime in terms of height, h get() O(h) put() O(h) remove() O(h) 5 fo 9 sho 15 sup What do you replace the node with? Largest in left sub tree or smallest in right sub tree burp poo boo CSE 373 SP 18 - KASEY CHAMPION 15

16 Binary Search Tree deleting a node 5 remove(15) // replace with CSE 373 AU 18 SHRI MARE 16

17 Binary Search Tree deleting a node 5 remove(15) // replace with 14 or CSE 373 AU 18 SHRI MARE 17