From 3bfa2429d4d80c32a2ef51e9bbe818e4c7a93ee4 Mon Sep 17 00:00:00 2001 From: Arkitu Date: Tue, 7 Oct 2025 14:38:27 +0200 Subject: [PATCH] td2 --- td2/td2.md | 38 ++++++++++++++++++++++++++++++++++++++ td2/td2.pdf | Bin 0 -> 162617 bytes 2 files changed, 38 insertions(+) create mode 100644 td2/td2.md create mode 100644 td2/td2.pdf diff --git a/td2/td2.md b/td2/td2.md new file mode 100644 index 0000000..0380eec --- /dev/null +++ b/td2/td2.md @@ -0,0 +1,38 @@ +# TD2 + +## Algorythme dichotomique en C + +```c +int f(int t[], int target, int len) { + int min = 0; + int max = len-1; + while (min < max) { + int i = (min+max)/2 + int val = t[i]; + if (target < val) { + max = i; + } else if (target > val) { + min = i; + } else { + min = i; + max = i; + } + } + return min; +} +``` + +Preuve de terminaison : + +- **variant :** $\lfloor log_2(\frac{max-min}{2}) \rfloor$ + +Preuve de correction : + +- **invariant :** $min \leq i_{target} \leq max$ +- **après la boucle :** $min = max$ + + donc $min \leq i_{target} \leq min$ + + donc $i_{target} = min$ + +Complexité : $O(log_2(n))$ diff --git a/td2/td2.pdf b/td2/td2.pdf new file mode 100644 index 0000000000000000000000000000000000000000..bb3194b461abb5e74b719c25786178b0c1194030 GIT binary patch literal 162617 zcma%?LzFHG5S+)lW81cE^Nwx&W7~7bwr$(CZQHh;mz*S=9Y%nVAu#53GIw5VR(6A7-RvqX3pk>OpJ`og#T+`7{o2CodJ%74C2;?&Hzz> zv7HG3hMynC$=MNLXanQEv8^Fzx5npWcBDlFt7dLt1>LcT5Z$pMe zs9k+fL>KY%iWmMMo>W{##d9_~3}QYrmC0cj0>;%&oYj?nH}kq|{qTfP9Ht;k4Qk>l z6_hlUk%no5B`Ier<0x*{+y7+bdhr62Vy=68Od1fhF50Hc@K{;!)zz+7yeO$9m%465 z4c<-9K1}^-zgbmc_3ESRk!4>3w=XOCwV#y~HmUFNSrT9_HGm9iii38KeOO?68U46) zBsxP3m!ZdggGTWx(ClS*Yy%LSZ9mCL^f7=N-3c1v?|cJ@o^j78ym@#eLl|4vP?@4PqmV8AsclG!s>EJZNm;hktM_AK&h|Cl?I}v)4~JH?btne1nj4 z3G`O&7BY+&{7pmIPj*5=Qv>k$|R7n!PoW7!C9PTmzyMAkIdi7Ad0?EprW(QCrc<L;0rd77XM!yLx%%yWe8OR+p~# zytZbkI^q}JnV^2{w0@%k&FlmY#YxUmsL2ca&RZeS*!L`EVVkHt)fnI0?`7!=Wwj7Tg0h~#K zoMDr$iGC?on<(u>5{dPSj>5B`acx;Xx}yG#< 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