Binary Search and BSTs .edu
CSE 373: Data Structures and Algorithms
Binary Search and BSTs
Autumn 2018
Shrirang (Shri) Mare shri@cs.washington.edu
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 ...
Administrivia
Due dates: - HW2 Part 2 due Friday 11:59pm - HW1 grades will be released later today
CSE 373 AU 18 ? SHRI MARE
2
Modeling recursion: Unfolding Method
nX2 T (n) = C2 + C1
i=0
AAACEHicbZDLSsNAFIYn9VbjrerSzWARK6Ik2eimUOzGZYXeIIlhMp22QyeTMDMRSugjuPFV3LhQxK1Ld76N0zYLbf1h4OM/53Dm/GHCqFSW9W0UVlbX1jeKm+bW9s7uXmn/oC3jVGDSwjGLRTdEkjDKSUtRxUg3EQRFISOdcFSf1jsPREga86YaJ8SP0IDTPsVIaSsonXqu2azwM1iFnkyjIKNVa3Kf8QtnUg8cCM9hPbBNzw9KZevSmgkug51DGeRqBKUvrxfjNCJcYYakdG0rUX6GhKKYkYnppZIkCI/QgLgaOYqI9LPZQRN4op0e7MdCP67gzP09kaFIynEU6s4IqaFcrE3N/2puqvrXfkZ5kirC8XxRP2VQxXCaDuxRQbBiYw0IC6r/CvEQCYSVztDUIdiLJy9D27m0Nd855dpNHkcRHIFjUAE2uAI1cAsaoAUweATP4BW8GU/Gi/FufMxbC0Y+cwj+yPj8AR/LmXY=
CSE 373 AU 18 ? SHRI MARE
3
Modeling binary search recursion
Question: Find the closed form for T(N)
( T (n) =
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
C1
n
C2 + T 2
when when
n=1 n>1
T (n) = C2 + T
AAACFnicbZDLSsNAFIYn9VbjrerSzWARWsSSdKMbodCNywq9QRPKZDpph04mYeZEKKFP4cZXceNCEbfizrdxello6w8DH/85hzPnDxLBNTjOt5Xb2Nza3snv2nv7B4dHheOTto5TRVmLxiJW3YBoJrhkLeAgWDdRjESBYJ1gXJ/VOw9MaR7LJkwS5kdkKHnIKQFj9QtXXs9ulmQZ3+J6v4ovcdMTLISSNwgVoZmcZtWpp/hwBGXb8/uFolNx5sLr4C6hiJZq9Atf3iCmacQkUEG07rlOAn5GFHAq2NT2Us0SQsdkyHoGJYmY9rP5WVN8YZwBDmNlngQ8d39PZCTSehIFpjMiMNKrtZn5X62XQnjjZ1wmKTBJF4vCVGCI8SwjPOCKURATA4Qqbv6K6YiYPMAkaZsQ3NWT16FdrbiG76vFmrOMI4/O0DkqIRddoxq6Qw3UQhQ9omf0it6sJ+vFerc+Fq05azlziv7I+vwB972cvg==
n 2
T (n) = C2 + C2 + T
AAACLnicbZBLSwMxEMezvl1fVY9egkWoCGW3CHoRCkXwqNCH0Cwlm2bb0Gx2SWaFsvQTefGr6EFQEa9+DNN2D74GQn75zwyT+YepFAY878VZWFxaXlldW3c3Nre2d0q7e22TZJrxFktkom9DargUirdAgOS3qeY0DiXvhKPGNN+549qIRDVhnPIgpgMlIsEoWKlXuiRdt1lRx/gCN3o1fIKJ5BFU5tycP0g/0pTlapKfTogWgyEcFxcmBLskcHulslf1ZoH/gl9AGRVx3Ss9kX7CspgrYJIa0/W9FIKcahBM8olLMsNTykZ0wLsWFY25CfLZuhN8ZJU+jhJtjwI8U7935DQ2ZhyHtjKmMDS/c1Pxv1w3g+g8yIVKM+CKzQdFmcSQ4Kl3uC80ZyDHFijTwv4VsyG13oB1eGqC/3vlv9CuVX3LN7Vy3SvsWEMH6BBVkI/OUB1doWvUQgzdo0f0it6cB+fZeXc+5qULTtGzj36E8/kFzNSkww==
n 4
T (n) = C2 + C2 + C2 + T
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
n 8
T (n) =
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
C| 2 + C2 {+z ? ? ? + C}2 t times
n + T 2t
We want to find a t such that
n 2t
AAACAXicbZDLSsNAFIZPvNZ4i7oR3AwWwVVJutGNUHDjsoK9QBLLZDpph04mYWYilFA3voobF4q49S3c+TZO2yy09YeBj/+cw5nzRxlnSrvut7Wyura+sVnZsrd3dvf2nYPDtkpzSWiLpDyV3QgrypmgLc00p91MUpxEnHai0fW03nmgUrFU3OlxRsMEDwSLGcHaWD3nOPDtoB9LTAoxKer3eoKukGcHYc+pujV3JrQMXglVKNXsOV9BPyV5QoUmHCvle26mwwJLzQinEzvIFc0wGeEB9Q0KnFAVFrMLJujMOH0Up9I8odHM/T1R4ESpcRKZzgTroVqsTc3/an6u48uwYCLLNRVkvijOOdIpmsaB+kxSovnYACaSmb8iMsQmDm1Cs00I3uLJy9Cu1zzDt/Vqwy3jqMAJnMI5eHABDbiBJrSAwCM8wyu8WU/Wi/VufcxbV6xy5gj+yPr8AV+YlXc=
=1
Solving for t we get
t AAAB+nicbZDLSsNAFIZP6q3GW6pLN4NFcFWSbnQjFNy4rGAv0IQwmU7aoZNJmJkoJfZR3LhQxK1P4s63cdpmoa0/DHz85xzOmT/KOFPadb+tysbm1vZOddfe2z84PHJqx12V5pLQDkl5KvsRVpQzQTuaaU77maQ4iTjtRZObeb33QKViqbjX04wGCR4JFjOCtbFCp+YPbI2uEU9HYRMJ2w9Cp+423IXQOngl1KFUO3S+/GFK8oQKTThWauC5mQ4KLDUjnM5sP1c0w2SCR3RgUOCEqqBYnD5D58YZojiV5gmNFu7viQInSk2TyHQmWI/Vam1u/lcb5Dq+CgomslxTQZaL4pwjnaJ5DmjIJCWaTw1gIpm5FZExlphok5ZtQvBWv7wO3WbDM3zXrLfcMo4qnMIZXIAHl9CCW2hDBwg8wjO8wpv1ZL1Y79bHsrVilTMn8EfW5w9syZIX
=
log2n
T (n)
AAACB3icbZBPS8MwGMbT+W/Wf1WPggSHMBFG24tehMEuHidsc9CWkmbpFpamJUmFUXbz4lfx4kERr34Fb34bs60H3Xwg8Mvzvi/J+0QZo1LZ9rdRWVvf2Nyqbps7u3v7B9bhUU+mucCki1OWin6EJGGUk66iipF+JghKIkbuo3FrVr9/IELSlHfUJCNBgoacxhQjpa3QOvU9s1PnF/AGtkIXQpYOQ5fDS31zTD8IrZrdsOeCq+CUUAOl2qH15Q9SnCeEK8yQlJ5jZyookFAUMzI1/VySDOExGhJPI0cJkUEx32MKz7UzgHEq9OEKzt3fEwVKpJwkke5MkBrJ5drM/K/m5Sq+DgrKs1wRjhcPxTmDKoWzUOCACoIVm2hAWFD9V4hHSCCsdHSmDsFZXnkVem7D0Xzn1pp2GUcVnIAzUAcOuAJNcAvaoAsweATP4BW8GU/Gi/FufCxaK0Y5cwz+yPj8AdzblWo=
=
C2log2n + C1
CSE 373 AU 18 ? SHRI MARE
4
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
5
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