From eb5ef07a693901bd1df624c4e2c1cfd5a3ef25f2 Mon Sep 17 00:00:00 2001 From: arthur Date: Tue, 11 Nov 2025 21:49:40 -0300 Subject: [PATCH] feat: new tutorial file, TLE solution updated --- .../knapsack-zero-one-tutorial.pdf | Bin 0 -> 106695 bytes .../knapsack-zero-one-tutorial.tex | 70 ++++++++++++++++++ knapsack-zero-one/knapsack-zero-one.pdf | Bin 158509 -> 84373 bytes knapsack-zero-one/problem.json | 4 +- knapsack-zero-one/src/TLE.cpp | 50 ++++++------- knapsack-zero-one/statement/tutorial.tex | 56 ++++++++++++++ 6 files changed, 151 insertions(+), 29 deletions(-) create mode 100644 knapsack-zero-one/knapsack-zero-one-tutorial.pdf create mode 100644 knapsack-zero-one/knapsack-zero-one-tutorial.tex diff --git a/knapsack-zero-one/knapsack-zero-one-tutorial.pdf b/knapsack-zero-one/knapsack-zero-one-tutorial.pdf new file mode 100644 index 0000000000000000000000000000000000000000..aca538b1bd4a101afc0919a81721e17eb84cf485 GIT binary patch literal 106695 zcma&NW2`Pf)2_K}?Pc4xZQHhO+iNe|Huth^+qP{z=bbt8C6m09%;_JUs&u-O%Khg` z^<89&qT+N+^lUI>^UEXaFl>a3gbv2mFuc4l460gadCAvGqQv6+&s~eir*4P@BPwvP7KXH*{nV%1tm?!lmw9l zX$K~-g)~ey%Vr_2Oh~eu9_Zt%XHT?E)wUkvhdU*XZt2!|NxP%_J~&-SwrK7qhEXDZ zUzy3KiAFoj(v_iTh&ZIH)%B$z@nO4cF@I~ZE5O&MIGo<}_aD_k+2hnW3CGwn#vID4sUk?%|-stzxx z5mf^{C)oE0N7+lQL{-`R{GcWTy_n)TxOWDQ#+*&8H|wPy;CxJ{#IqK9#*X|_e6X>h z1#c0Q?{PC4TLmXTWR#rUVxzi>DZrJvprGF7?>-GZZ!@@CtHb;Ex0c0@H{xOK&(sy8 zY(fHD-AP>&o{LZ*-gmcKKy}^YThjVY%skvWk_Rz7>((zKX7knNX01YGO^T!mz80Y( zHC6Hc=4i7Ja?3?6lDkREKhZiEb!$>C$W&xin2WZ)!ITr-1-Q@w%!=t{k+xW8M-lZX zTPg=0sMSO9@*lgLjBi)Z zU5Q6R#hElEstMde3nBij^^6C8K0z3K`I#knQI^B57hJq;#1GZ$#%LoIp~>J|R;|#& zV4&U=?Z;-(P|C_<;C8`aA~_4~`)<*pg27Skha$mXf*V0Vlntu0V*3yLZABZL0ikQR z1^JgITVZUo&DZ%GJ1rh>%$;3io?SNC4aG2X-scqeh5w>$%9ekCk!!booabHKj-($asl-6P24$e-DIKa17lQYQp7Gvu&+?A^>%Ges7= zD=r-aJG7(uq$dQ@CYF5q>rp!?zS7V(sAQe|DKNuEcI^E?$mGvR$4;@59 zVi@t;CZjc=WzKuMh8*k05<@yCxqE}&#kF!zV-xv1PG-~2+wpoWlreM+zI9DPi?9Xu zm;WW@B*|HbG-yi(f7cICgtpA@9y>1Lf<+QR2T(nEmaZAX-*hj zHZ4;GUK*>U4#e`mdWi}XNWUV((Yi}HyTOm(uZ53sIZt|Han1ds7OX9!X5}(goAO*x z^3KSmHL5U^AnCHXqfURux+yekmQr(tek(h=f<)2*0(s)dPShWzNq^4v^oCi`Ytp`L zU;nt*!GAKHoTh!{fxy0^0^|CjKMjg0)9w=*a>IGN_s7Asb_mW zV9f#XB%ydMk!kK2Gbd-ca&@!fYZjm~r&Ps(kHJ^4vBW@$a=szW7*;`ke9nqKDv{q$ z_tH|ZZ>{xmgx-V9k;;*j&JYHLoIK?mf*Fh8Ga$ObGRR8{b zw852%`R%^0lSJp zifF!y!9tfd-F;IxGNVNSDcf>@v#?osQ|XFkz??OtEY;)LGq`|$k`3K(vH2x;(u8dX6)W7x1!Bg=} z_m|Z3iPtOsXh*nof{95renwD(BRkzp0+Q1HeOnZM11RQmef6VfN0cdhI6ppv0v3dZWuz{TA25SdQVryj zsNj+#`E~yqdJB`#O|~5SXO&sb_NtxWjgTF%!^7@;QIlQ=+|ImEM_6nG%y8mr>-gCY zT&raAdBlJQ9oU0Tv?0c8W4l{fCNk6B`BL3qFX0UUx%HAz5|iEoK?`_n#a`@t|LuRd zz3>67zcRi?IH0@R>E(*W@7Y4p94I?P{M+ehMAR3`Dti;Fe<1PMa{d?Kevll$QIb_G z3gn(61<7g323c$=OS+}@)7AegT1Vu!#d>{>()_{<>1n$qXeN$s$*`6gqv>vLxyg=J zJSBeG(eOZ=dDRik=0vNL5aI&jT>dB^^e64a@2#bwFpblVk%MNGaAx;nYK!`N!($fu zS{Bvc5#oy@Gr~A-e4E-Qk#;KVCuwrc?{y`ppMV;m$(`lbx78jYbmIMNzo?;a_;=FH zP*GoUf>3%Cd{)*%KD+1Yh{kWYr;eRXcGU0SU+rk_o&5$wk&2=HrMtGMB8OEOwr&Yb zz2CaPT1NF;o&Q+noJDn&%RdJ8+wq07U6D+iJ{WWm!9w`>T>)s}o|x0y5HvEI@CZGE zIQJgS>!K^Y8=O9Co3aD5ugyN+FTDcSES;vX6$XJ&cq6|GqHA5D>+2Q*0l4=y;}PZr z&ufA^y4OL==W_{x9o|n^1Jpb&7&Cj*{|7Ao6a8OM@*fWQFFIjiW&7W7LiqnrCw58y z>_Zrl!ryqqBy_6q@?-y1E|)5^xLyP=2QhOFs3l4ka7r2Uv4^v!YI1(NyzT77=TFVJ zr-M>!CdS~V=0gcGCrgzCvCy+o~dA&(qKB7U-!T?8tGzeEdw_sJZ*7;itrz*Zv zg;6SJ94of%K>M$SnrC4JRY0LKMm>sKj&kAS6$bv4k_67RUKGQ;hlsd0;uR`SIU*&Q zY8$3YXX$JTR?N2(oTWYbGfZg3y)!>dDjFG4PA4hXGJ-92=p)P1K#lwYy> z^zw(&AICl*y@JH{G0@X-b{e_v=0-9O)#{U+IiIwaBSSS{zWfnGNMny(& zVG|Z`6PGYlkvd^|6Hss@R(22>(lci+FXxL08cZC=3m+h{j%PkUH0fw*m5%3JkQ_9) zj%Pj8B6u`7;|^XuRLo-nM#gWzQ2#rC0Q3t0fd2*n2oeAQK+w;B0>Sp*|Lp%c2LJ?y z4NdtNT?Ea8ry(|g8(~c;#_j1f6zT-&fks{2O#iz`*8eV&iI9ntjro7bnVFD@k%NQd ze^M4g#{UTknV7g(nf?!%I<^^H5qBM(4u+_`OXTSG_7)j8m}6d0#3a^d9t=7G3<`Oh zuf40wCU75#mvaC=Gn0qq?bqN%-%ZVI zRx>aLM__VtL~e32PFS2gidi# za3Kl0pNy2W|5Jifa1s~{Tn7XX*jkW7Ct=7%OokAw;KmdhqQeWI0VA%Y!@*%;gn*!+ zpx|B*Leqt^ES?jEwh!&z1a29mE2M~+fVTfO4&+t-Ddc4x!=Jps3V1TKZ$kxPNDUp07A20KLqUf&*dlgw*GuCsy{9O z43v&87dIy;Zr4x(Cy)&j1f)Aj!_4E!*#sO2uht(W2uY*-%(DOz2@Y7B7vp zZFgZ}Y2i&^9Q)eR?8gU=#=`7Y7ih)YS!| zX(+|%2QdY87yY34YQi%8q44hwpj?MF9R8pcpa(j;o7)4(=AoIN978?)M*NPUxmZqC zT2EXyeX~98$y-@bFa$E-5FiG|X2-+!kBpFjA?+W5y#qY(;R)aC`8femYR=8>AW%OW zw@+L@AKxP{0NOVJ;a`C2FV7S}?HsU` z*v$j>7?J;=FbEh!bOjRW1{m|wtnJ)*!7k_Ic?>)RRDKo+XlQIg8^84NlX2`n(Mxh| z3jXqoH&pYk!`{I!8>b#rn6M?SmsGGAwBLoZs% z0qufwQ%`$jC%AOEHU17*90USDP+=fFN}Tzn!!u)$_b1-^5;O^G1k@0Y4LsZ(d=Wr- z5l@hsJVHwTdz1nKfg+!67Yd|6`w1VA&>)!o){zOQD0 zP&W`lZeQZO3;4dkc}MVwfFxAE(Fez&e_RuoRuEA_z(c@RzoLu>q|Nw~E`SNZ&xueS zAVd!Yc}_46zuy1>fdB9pq3}5GJNE@bxbNF{QAr3GIIF|`4bq(}#RY)p%W(a`d*ye` z&HeOk&yC~&oCFA3pPEi{e^+SMo+JU6VOJd(YyY_d)fS)~K!0iu3;yHNx(CKMd`Ym> zq|CX^oBNx3^fA5z2n3Yk%6L#dfBbx#wB?(<5PXlBz=L=N(JG;z<wYA`BO2zm2ts z57uyVcxX$!@35Q6h4DvQ{yL;;ed}Bp6#{2*)k3#w5?zL}w>_4e<}BCGU`f|l;>0>q zNSP$iRQyKVdpiEWu5%jOp8c#zxpeGuM_NM0K~5afJNzrV_$7TuMvXH2g?>6l9U*a| zlbk)f64Pc3qUT9dxrreb6EU_H5#^r&K7ndtFCRXpZEZg

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+\end{verbatim} + +O resultado final é dado por $dp[n][W]$, que representa o maior valor possível que pode ser obtido sem ultrapassar a capacidade da mochila. +\end{document} diff --git a/knapsack-zero-one/knapsack-zero-one.pdf b/knapsack-zero-one/knapsack-zero-one.pdf index 1f9a2b66b09bcab328f6b8b28cd7cee366f06ae1..f8f2564d38eaad6c69c55a95e8e8eac214b2a3f7 100644 GIT binary patch delta 51225 zcmb5WWk3{K7dAS~Ff)BL(z)VX4Co%(}yeZn)Y)Kjz@X_j}q>9)0Z|RiR zYx?i$K3#m2ysvkf*dLTGOzSJ}E~669*%r_ebSq-Ya?R(2F*#0Q^*;Y@0o6+Pc`%uz zxxcs1-n1miZB+XpbZt{}G7VAyn>deo*P(KX++MZWW(;&w$BeV4k+< z#BDbBJ$!m6L(=TR%?G~M8wMD(gi%Zkp;|nckNF(%rJHn3k|c*6{};vUc^RtiSpD5r z&QGQnz8#ozvZJAfq%wTP1gno2(yZsdb-Itl)?@x`JD&co<*Xl_fdIYzkluE<*kZQG zs8#a}do=hpB`(Hjqj$HlFlnmFFT`a1a6x#UbjHUM{v|)ZnAQyV;>-(ySiR`4UfT|# zY#;J4%QerbwyFmkT&h#=KGD^NJ8r|LSSxTIm$bj6GTS5Yg=V^vN`9B~R4HicO_`cRMws?-TZH zki~-e;&jP)Tbz3ea9?AHNiyz{!`@ z4#KT5vVjn~ct>t8a@=T$xK-z|tna(-@rqUzi>EFu12oZut+??KxWb?l@rI%~FGn1H z_~drUWzm~*mlx`Fl|2OBAc)a>y-0gRwg_IfmPmacl 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a/knapsack-zero-one/problem.json b/knapsack-zero-one/problem.json index feed1c4..f8c7b63 100644 --- a/knapsack-zero-one/problem.json +++ b/knapsack-zero-one/problem.json @@ -11,10 +11,10 @@ "grader": false, "subject": { "en_us": [ - "dynamic-programming" + "dynamic-programming", "knapsack-problem", "knapsack-0/1" ], "pt_br": [ - "programação-dinâmica" + "programação-dinâmica", "problema-da-mochila", "problema-da-mochila-booleano" ], "es": [ "" diff --git a/knapsack-zero-one/src/TLE.cpp b/knapsack-zero-one/src/TLE.cpp index 2ff9624..d4b5f32 100644 --- a/knapsack-zero-one/src/TLE.cpp +++ b/knapsack-zero-one/src/TLE.cpp @@ -1,34 +1,30 @@ #include using namespace std; -typedef long long ll; +using ll = long long; + +int N; +ll W; +vector weight, value; + +ll knapsack(int i, ll w) { + if (i == 0 || w == 0) return 0; + + ll res = knapsack(i - 1, w); + if (weight[i] <= w) + res = max(res, value[i] + knapsack(i - 1, w - weight[i])); + + return res; +} + int main() { - int N, W; cin >> N >> W; - vector> items(N+1); - for (int i = 1; i <= N; i++) { - ll wi, vi; cin >> wi >> vi; - items[i] = {wi, vi}; - } + cin >> N >> W; + weight.resize(N + 1); + value.resize(N + 1); - ll ans = 0; - for (ll i = 0; i < (1<> j) & 1)) { - if (capacity >= wj) { - capacity -= wj; - value += vj; - } else { - value = -1; - break; - } - } - } - ans = max(ans, value); - } - - cout << ans << endl; + for (int i = 1; i <= N; i++) + cin >> weight[i] >> value[i]; + cout << knapsack(N, W) << "\n"; return 0; -} \ No newline at end of file +} diff --git a/knapsack-zero-one/statement/tutorial.tex b/knapsack-zero-one/statement/tutorial.tex index e69de29..32a5ebf 100644 --- a/knapsack-zero-one/statement/tutorial.tex +++ b/knapsack-zero-one/statement/tutorial.tex @@ -0,0 +1,56 @@ +\section{Solução do Problema} + +A solução do problema da mochila 0/1 pode ser abordada por meio de \textit{programação dinâmica}. +A ideia central é decompor o problema em subproblemas menores, de forma que a solução ótima total seja composta pelas soluções ótimas desses subproblemas. + +\subsection{Definição do Subproblema} + +Seja $dp[i][w]$ o valor máximo que pode ser obtido ao considerar os $i$ primeiros itens, com capacidade máxima da mochila igual a $w$. + +\subsection{Função de Transição} + +Para cada item $i$ (com peso $p_i$ e valor $v_i$), temos duas opções: + +\begin{itemize} + \item Não incluir o item $i$: o valor máximo permanece igual ao do subproblema anterior, isto é, $dp[i-1][w]$. + \item Incluir o item $i$: caso o peso $p_i$ caiba na mochila ($p_i \leq w$), o valor máximo será o valor do item $v_i$ somado ao melhor valor possível com a capacidade restante, $dp[i-1][w - p_i]$. +\end{itemize} + +Assim, a função de transição pode ser expressa como: + +\[ +dp[i][w] = +\begin{cases} +dp[i-1][w], & \text{se } p_i > w \\ +\max(dp[i-1][w],\; dp[i-1][w - p_i] + v_i), & \text{caso contrário} +\end{cases} +\] + +\subsection{Casos Base} + +As condições iniciais são: + +\[ +dp[0][w] = 0, \quad \forall w \geq 0 +\] +\[ +dp[i][0] = 0, \quad \forall i \geq 0 +\] + +Esses casos representam que, com zero itens ou capacidade zero, o valor máximo obtido é zero. + +\subsection{Implementação Dinâmica} + +O algoritmo preenche uma tabela $dp$ de tamanho $(n+1) \times (W+1)$, onde $n$ é o número de itens e $W$ é a capacidade máxima da mochila. +Cada célula é calculada a partir das decisões descritas na função de transição. + +\begin{verbatim} +for i in 1..n: + for w in 1..W: + if peso[i] <= w: + dp[i][w] = max(dp[i-1][w], dp[i-1][w - peso[i]] + valor[i]) + else: + dp[i][w] = dp[i-1][w] +\end{verbatim} + +O resultado final é dado por $dp[n][W]$, que representa o maior valor possível que pode ser obtido sem ultrapassar a capacidade da mochila.