pythonread.github.io/dsa/dynamic-programming.html at main · pythonread/pythonread.github.io

History

771 lines (521 loc) · 29.2 KB

Raw

100

101

102

103

104

105

106

107

108

109

110

111

112

113

114

115

116

117

118

119

120

121

122

123

124

125

126

127

128

129

130

131

132

133

134

135

136

137

138

139

140

141

142

143

144

145

146

147

148

149

150

151

152

153

154

155

156

157

158

159

160

161

162

163

164

165

166

167

168

169

170

171

172

173

174

175

176

177

178

179

180

181

182

183

184

185

186

187

188

189

190

191

192

193

194

195

196

197

198

199

200

201

202

203

204

205

206

207

208

209

210

211

212

213

214

215

216

217

218

219

220

221

222

223

224

225

226

227

228

229

230

231

232

233

234

235

236

237

238

239

240

241

242

243

244

245

246

247

248

249

250

251

252

253

254

255

256

257

258

259

260

261

262

263

264

265

266

267

268

269

270

271

272

273

274

275

276

277

278

279

280

281

282

283

284

285

286

287

288

289

290

291

292

293

294

295

296

297

298

299

300

301

302

303

304

305

306

307

308

309

310

311

312

313

314

315

316

317

318

319

320

321

322

323

324

325

326

327

328

329

330

331

332

333

334

335

336

337

338

339

340

341

342

343

344

345

346

347

348

349

350

351

352

353

354

355

356

357

358

359

360

361

362

363

364

365

366

367

368

369

370

371

372

373

374

375

376

377

378

379

380

381

382

383

384

385

386

387

388

389

390

391

392

393

394

395

396

397

398

399

400

401

402

403

404

405

406

407

408

409

410

411

412

413

414

415

416

417

418

419

420

421

422

423

424

425

426

427

428

429

430

431

432

433

434

435

436

437

438

439

440

441

442

443

444

445

446

447

448

449

450

451

452

453

454

455

456

457

458

459

460

461

462

463

464

465

466

467

468

469

470

471

472

473

474

475

476

477

478

479

480

481

482

483

484

485

486

487

488

489

490

491

492

493

494

495

496

497

498

499

500

501

502

503

504

505

506

507

508

509

510

511

512

513

514

515

516

517

518

519

520

521

522

523

524

525

526

527

528

529

530

531

532

533

534

535

536

537

538

539

540

541

542

543

544

545

546

547

548

549

550

551

552

553

554

555

556

557

558

559

560

561

562

563

564

565

566

567

568

569

570

571

572

573

574

575

576

577

578

579

580

581

582

583

584

585

586

587

588

589

590

591

592

593

594

595

596

597

598

599

600

601

602

603

604

605

606

607

608

609

610

611

612

613

614

615

616

617

618

619

620

621

622

623

624

625

626

627

628

629

630

631

632

633

634

635

636

637

638

639

640

641

642

643

644

645

646

647

648

649

650

651

652

653

654

655

656

657

658

659

660

661

662

663

664

665

666

667

668

669

670

671

672

673

674

675

676

677

678

679

680

681

682

683

684

685

686

687

688

689

690

691

692

693

694

695

696

697

698

699

700

701

702

703

704

705

706

707

708

709

710

711

712

713

714

715

716

717

718

719

720

721

722

723

724

725

726

727

728

729

730

731

732

733

734

735

736

737

738

739

740

741

742

743

744

745

746

747

748

749

750

751

752

753

754

755

756

757

758

759

760

761

762

763

764

765

766

767

768

769

770

771

<!DOCTYPE html>

<head>

<title>Dynamic Programming</title>

window.dataLayer = window.dataLayer || [];

function gtag() { dataLayer.push(arguments); }

gtag('js', new Date());

gtag('config', 'G-HCERBDV76D');

</script>

</head>

<body>

<nav>

Python Tutorial

</a>

</div>

<span>

</span>

<span>

</span>

<span>

</span>

</div>

</div>

<li>

Home

</a>

</li>

<li>

Projects

</a>

</li>

<li>

Free Course

</a>

</li>

<li>

DSA

</a>

</li>

</ul>

</div>

</nav>

</div>

<h3>Page Index</h3>

<ul>

<li><a href="/dsa.html#data-structure-1" title="Data Structures (I)">Data Structures

(I)</a></li>

<li><a href="/dsa.html#data-structure-2" title="Data Structures (II)">Data Structures

(II)</a></li>

<li><a href="/dsa.html#tree-1" title="Tree based DSA (I)">Tree based DSA (I)</a></li>

<li><a href="/dsa.html#tree-2" title="Tree based DSA (II)">Tree based DSA (II)</a></li>

<li><a href="/dsa.html#graph" title="Graph Data Structures and Algorithm">Graph based

DSA</a></li>

<li><a href="/dsa.html#sorting-searching" title="Sorting and Searching">Sorting and

Searching</a></li>

<li><a href="/dsa.html#greedy-algorithm" title="Greedy Algorithms">Greedy Algorithms</a>

</li>

<li><a href="/dsa.html#dynamic-programming" title="Dynamic Programming">Dynamic

Programming</a></li>

<li><a href="/dsa.html#other-algorithms" title="Other Algorithms">Other Algorithms</a>

</li>

</ul>

</div>

<h1>Dynamic Programming</h1>

<p class="editor-contents__short-description">In this tutorial, you will learn what dynamic programming is. Also, you will find the comparison between dynamic programming and greedy algorithms to solve problems.</p>

<p id="definition">Dynamic Programming is a technique in computer programming that helps to efficiently solve a class of problems that have overlapping subproblems and <a href="https://en.wikipedia.org/wiki/Optimal_substructure" target="_blank">optimal substructure</a> property.</p>

<p>If any problem can be divided into subproblems, which in turn are divided into smaller subproblems, and if there are overlapping among these subproblems, then the solutions to these subproblems can be saved for future reference. In this way, efficiency of the CPU can be enhanced. This method of solving a solution is referred to as dynamic programming.</p>

<p>Such problems involve repeatedly calculating the value of the same subproblems to find the optimum solution.</p>

<hr><h2 id="example">Dynamic Programming Example</h2>

<p>Let's find the fibonacci sequence upto 5th term. A fibonacci series is the sequence of numbers in which each number is the sum of the two preceding ones. For example, <code>0,1,1, 2, 3</code>. Here, each number is the sum of the two preceding numbers.</p>

<p><strong>Algorithm</strong></p>

<pre>Let n be the number of terms.

1. If n <= 1, return 1.

2. Else, return the sum of two preceding numbers.

</pre>

<p>We are calculating the fibonacci sequence up to the 5th term.</p>

<ol><li>The first term is 0.</li>

<li>The second term is 1.</li>

<li>The third term is sum of 0 (from step 1) and 1(from step 2), which is 1.</li>

<li>The fourth term is the sum of the third term (from step 3) and second term (from step 2) i.e. <code>1 + 1 = 2</code>.</li>

<li>The fifth term is the sum of the fourth term (from step 4) and third term (from step 3) i.e. <code>2 + 1 = 3</code>.</li>

</ol><p>Hence, we have the sequence <code>0,1,1, 2, 3</code>. Here, we have used the results of the previous steps as shown below. This is called a <strong>dynamic programming approach</strong>.</p>

<pre>F(0) = 0

F(1) = 1

F(2) = F(1) + F(0)

F(3) = F(2) + F(1)

F(4) = F(3) + F(2)</pre>

<hr><h2 id="working">How Dynamic Programming Works</h2>

<p>Dynamic programming works by storing the result of subproblems so that when their solutions are required, they are at hand and we do not need to recalculate them.</p>

<p>This technique of storing the value of subproblems is called memoization. By saving the values in the array, we save time for computations of sub-problems we have already come across.</p>

<pre>var m = map(0 → 0, 1 → 1)

function fib(n)

if key n is not in map m

m[n] = fib(n − 1) + fib(n − 2)

return m[n]</pre>

<p></p><div class="clearfix"></div><p>Dynamic programming by memoization is a top-down approach to dynamic programming. By reversing the direction in which the algorithm works i.e. by starting from the base case and working towards the solution, we can also implement dynamic programming in a bottom-up manner.</p>

<pre>function fib(n)

if n = 0

return 0

else

var prevFib = 0, currFib = 1

repeat n − 1 times

var newFib = prevFib + currFib

prevFib = currFib

currFib = newFib

return currentFib</pre>

<hr><h2 id="comparison-1">Recursion vs Dynamic Programming</h2>

<p>Dynamic programming is mostly applied to recursive algorithms. This is not a coincidence, most optimization problems require recursion and dynamic programming is used for optimization.</p>

<p>But not all problems that use recursion can use Dynamic Programming. Unless there is a presence of overlapping subproblems like in the fibonacci sequence problem, a recursion can only reach the solution using a divide and conquer approach.</p>

<p>That is the reason why a recursive algorithm like Merge Sort cannot use Dynamic Programming, because the subproblems are not overlapping in any way.</p>

<hr><h2 id="comparison-2">Greedy Algorithms vs Dynamic Programming</h2>

<p><a href="/dsa/greedy-algorithm.html">Greedy Algorithms</a> are similar to dynamic programming in the sense that they are both tools for optimization.</p>

<p>However, greedy algorithms look for locally optimum solutions or in other words, a greedy choice, in the hopes of finding a global optimum. Hence greedy algorithms can make a guess that looks optimum at the time but becomes costly down the line and do not guarantee a globally optimum.</p>

<p>Dynamic programming, on the other hand, finds the optimal solution to subproblems and then makes an informed choice to combine the results of those subproblems to find the most optimum solution.</p>

<hr><h2 id="algorithms">Different Types of Dynamic Programming Algorithms</h2>

<ol><li><a href="/dsa/longest-common-subsequence.html">Longest Common Subsequence</a></li>

<li><a href="/dsa/floyd-warshall-algorithm.html">Floyd-Warshall Algorithm</a></li>

</ol></div>

</div>

<div class="tutorial-toc"><div class="tutorial-toc__inner"><h3 class="tutorial-toc__title">Table of Contents

<li><a href="#example">Dynamic Programming Example</a></li>

<li><a href="#working">How Dynamic Programming Works</a></li>

<li><a href="#comparison-1">Recursion vs Dynamic Programming</a></li>

<li><a href="#comparison-2">Greedy Algorithms vs Dynamic Programming</a></li>

<li><a href="#algorithms">Different Types of Greedy Algorithm</a></li>

</ul></div></div></div> </div>

<div>

<ul>

<button style="padding:5px; margin-bottom:2px"><a href="

/dsa/stack.html">

Stack</a></button>

<button style="padding:5px; margin-bottom:2px"><a href="

/dsa/queue.html">

Queue</a></button>

<button style="padding:5px; margin-bottom:2px"><a href="

/dsa/types-of-queue.html">

Types of Queue</a></button>

<button style="padding:5px; margin-bottom:2px"><a href="

/dsa/circular-queue.html">

Circular Queue</a></button>

<button style="padding:5px; margin-bottom:2px"><a href="

/dsa/priority-queue.html">

Priority Queue</a></button>

<button style="padding:5px; margin-bottom:2px"><a href="

/dsa/deque.html">

Deque</a></button>

<br/>

<button style="padding:5px; margin-bottom:2px"><a href="

/dsa/linked-list.html">

Linked List</a></button>

<button style="padding:5px; margin-bottom:2px"><a href="

/dsa/linked-list-operations.html">

Linked List Operations</a></button>

<button style="padding:5px; margin-bottom:2px"><a href="

/dsa/linked-list-types.html">

Types of Linked List</a></button>

<button style="padding:5px; margin-bottom:2px"><a href="

/dsa/hash-table.html">

Hash Table</a></button>

<button style="padding:5px; margin-bottom:2px"><a href="

/dsa/heap-data-structure.html">

Heap Data Structure</a></button>

<button style="padding:5px; margin-bottom:2px"><a href="

/dsa/fibonacci-heap.html">

Fibonacci Heap</a></button>

<button style="padding:5px; margin-bottom:2px"><a href="

/dsa/decrease-key-and-delete-node-from-a-fibonacci-heap.html">

Decrease Key and Delete node from Fibonacci Heap</a></button>

<br>

<br/>

<button style="padding:5px; margin-bottom:2px"><a href="

/dsa/trees.html">

Tree Data Structure</a></button>

<button style="padding:5px; margin-bottom:2px"><a href="

/dsa/tree-traversal.html">

Tree Traversal</a></button>

<button style="padding:5px; margin-bottom:2px"><a href="

/dsa/binary-tree.html">

Binary Tree</a></button>

<button style="padding:5px; margin-bottom:2px"><a href="

/dsa/full-binary-tree.html">

Full Binary Tree</a></button>

<button style="padding:5px; margin-bottom:2px"><a href="

/dsa/perfect-binary-tree.html">

Perfect Binary Tree</a></button>

<button style="padding:5px; margin-bottom:2px"><a href="

/dsa/complete-binary-tree.html">

Complete Binary Tree</a></button>

<button style="padding:5px; margin-bottom:2px"><a href="

/dsa/balanced-binary-tree.html">

Balanced Binary Tree</a></button>

<button style="padding:5px; margin-bottom:2px"><a href="

/dsa/binary-search-tree.html">

Binary Search Tree</a></button>

<button style="padding:5px; margin-bottom:2px"><a href="

/dsa/avl-tree.html">

AVL Tree</a></button>

<br>

<br/>

<button style="padding:5px; margin-bottom:2px"><a href="

/dsa/b-tree.html">

B Tree</a></button>

<button style="padding:5px; margin-bottom:2px"><a href="

/dsa/insertion-into-a-b-tree.html">

Insertion into B-tree</a></button>

<button style="padding:5px; margin-bottom:2px"><a href="

/dsa/deletion-from-a-b-tree.html">

Deletion from B-tree</a></button>

<button style="padding:5px; margin-bottom:2px"><a href="

/dsa/b-plus-tree.html">

B+ Tree</a></button>

<button style="padding:5px; margin-bottom:2px"><a href="

/dsa/insertion-on-a-b-plus-tree.html">

Insertion on a B+ Tree</a></button>

<button style="padding:5px; margin-bottom:2px"><a href="

/dsa/deletion-from-a-b-plus-tree.html">

Deletion from a B+ Tree</a></button>

<button style="padding:5px; margin-bottom:2px"><a href="

/dsa/red-black-tree.html">

Red Black Tree</a></button>

<button style="padding:5px; margin-bottom:2px"><a href="

/dsa/insertion-in-a-red-black-tree.html">

Insertion in Red Black Tree</a></button>

<button style="padding:5px; margin-bottom:2px"><a href="

/dsa/deletion-from-a-red-black-tree.html">

Deletion from Red Black Tree</a></button>

<br>

<br/>

<button style="padding:5px; margin-bottom:2px"><a href="

/dsa/graph.html">

Graph Data Structure</a></button>

<button style="padding:5px; margin-bottom:2px"><a href="

/dsa/spanning-tree-and-minimum-spanning-tree.html">

Spanning Tree</a></button>

<button style="padding:5px; margin-bottom:2px"><a href="

/dsa/strongly-connected-components.html">

Strongly Connected Components</a></button>

<button style="padding:5px; margin-bottom:2px"><a href="

/dsa/graph-adjacency-matrix.html">

Adjacency Matrix</a></button>

<button style="padding:5px; margin-bottom:2px"><a href="

/dsa/graph-adjacency-list.html">

Adjacency List</a></button>

<button style="padding:5px; margin-bottom:2px"><a href="

/dsa/graph-dfs.html">

DFS Algorithm</a></button>

<button style="padding:5px; margin-bottom:2px"><a href="

/dsa/graph-bfs.html">

Breadth-first Search</a></button>

<button style="padding:5px; margin-bottom:2px"><a href="

/dsa/bellman-ford-algorithm.html">

Bellman Ford's Algorithm</a></button>

<br>

<br/>

<button style="padding:5px; margin-bottom:2px"><a href="

/dsa/bubble-sort.html">

Bubble Sort</a></button>

<button style="padding:5px; margin-bottom:2px"><a href="

/dsa/selection-sort.html">

Selection Sort</a></button>

<button style="padding:5px; margin-bottom:2px"><a href="

/dsa/insertion-sort.html">

Insertion Sort</a></button>

<button style="padding:5px; margin-bottom:2px"><a href="

/dsa/merge-sort.html">

Merge Sort</a></button>

<button style="padding:5px; margin-bottom:2px"><a href="

/dsa/quick-sort.html">

Quick Sort</a></button>

<button style="padding:5px; margin-bottom:2px"><a href="

/dsa/counting-sort.html">

Counting Sort</a></button>

<button style="padding:5px; margin-bottom:2px"><a href="

/dsa/radix-sort.html">

Radix Sort</a></button>

<button style="padding:5px; margin-bottom:2px"><a href="

/dsa/bucket-sort.html">

Bucket Sort</a></button>

<button style="padding:5px; margin-bottom:2px"><a href="

/dsa/heap-sort.html">

Heap Sort</a></button>

<button style="padding:5px; margin-bottom:2px"><a href="

/dsa/shell-sort.html">

Shell Sort</a></button>

<button style="padding:5px; margin-bottom:2px"><a href="

/dsa/linear-search.html">

Linear Search</a></button>

<button style="padding:5px; margin-bottom:2px"><a href="

/dsa/binary-search.html">

Binary Search</a></button>

<br>

<br/>

<button style="padding:5px; margin-bottom:2px"><a href="

/dsa/greedy-algorithm.html">

Greedy Algorithm</a></button>

<button style="padding:5px; margin-bottom:2px"><a href="

/dsa/ford-fulkerson-algorithm.html">

Ford-Fulkerson Algorithm</a></button>

<button style="padding:5px; margin-bottom:2px"><a href="

/dsa/dijkstra-algorithm.html">

Dijkstra's Algorithm</a></button>

<button style="padding:5px; margin-bottom:2px"><a href="

/dsa/kruskal-algorithm.html">

Kruskal's Algorithm</a></button>

<button style="padding:5px; margin-bottom:2px"><a href="

/dsa/prim-algorithm.html">

Prim's Algorithm</a></button>

<button style="padding:5px; margin-bottom:2px"><a href="

/dsa/huffman-coding.html">

Huffman Code</a></button>

<br>

<br/>

<button style="padding:5px; margin-bottom:2px"><a href="

/dsa/dynamic-programming.html">

Dynamic Programming</a></button>

<button style="padding:5px; margin-bottom:2px"><a href="

/dsa/floyd-warshall-algorithm.html">

Floyd Warshall Algorithm</a></button>

<button style="padding:5px; margin-bottom:2px"><a href="

/dsa/longest-common-subsequence.html">

Longest Common Subsequence</a></button>

<br>

<br/>

<button style="padding:5px; margin-bottom:2px"><a href="

/dsa/backtracking-algorithm.html">

Backtracking Algorithm</a></button>

<button style="padding:5px; margin-bottom:2px"><a href="

/dsa/rabin-karp-algorithm.html">

Rabin-Karp Algorithm</a></button>

<br>

<br/>

<h2>Free Courses on YouTube</h2>

<br/>

<button style="padding:5px; margin-bottom:2px"><a href="

https://www.youtube.com/watch?v=IbSXF7eT-AU&list=PLR_5PTwg_uAQw40OhdvAwN4NvzWv0xOZ2">

Python Full Course Playlist</a></button>

<br/>

105 STL Algorithms in Less Than an Hour</a></button>

<br/>

C++ STL Playlist</a></button>

<br/>

Learn Node.js </a></button>

<br/>

Learn Data Science Full course</a></button>

<br/>

Learn Computer Networking Full course</a></button>

<br/>

</br></br></br></br></br></br></br></br></ul>

</div>

<p>

<a href="/cookie-policy.html">Cookie policy</a> |

<a href="/privacy-policy.html">Privacy policy</a> |

<a href="/terms-of-use.html">Terms of use</a> |

<a href="/disclaimer.html">Disclaimer</a> |

<a href="/about-us.html">About Us</a>

</p>

<p>

</p>

</footer>

</script>

function changepy(){

document.getElementById('py1').className = 'tabbed-editor__node tabbed-editor__node--active';

document.getElementById('java1').className = 'tabbed-editor__node';

document.getElementById('c1').className = 'tabbed-editor__node';

document.getElementById('cpp1').className = 'tabbed-editor__node';

document.getElementById('python-code').className = 'code-editor__area code-editor__area--active';

document.getElementById('java-code').className = 'code-editor__area';

document.getElementById('c-code').className = 'code-editor__area';

document.getElementById('cpp-code').className = 'code-editor__area';

}

function changecpp(){

document.getElementById('py1').className = 'tabbed-editor__node';

document.getElementById('java1').className = 'tabbed-editor__node';

document.getElementById('c1').className = 'tabbed-editor__node';

document.getElementById('cpp1').className = 'tabbed-editor__node tabbed-editor__node--active';

document.getElementById('python-code').className = 'code-editor__area';

document.getElementById('java-code').className = 'code-editor__area';

document.getElementById('c-code').className = 'code-editor__area';

document.getElementById('cpp-code').className = 'code-editor__area code-editor__area--active';

}

function changejava(){

document.getElementById('py1').className = 'tabbed-editor__node';

document.getElementById('java1').className = 'tabbed-editor__node tabbed-editor__node--active';

document.getElementById('c1').className = 'tabbed-editor__node';

document.getElementById('cpp1').className = 'tabbed-editor__node';

document.getElementById('python-code').className = 'code-editor__area';

document.getElementById('java-code').className = 'code-editor__area code-editor__area--active';

document.getElementById('c-code').className = 'code-editor__area';

document.getElementById('cpp-code').className = 'code-editor__area';

}

function changec(){

document.getElementById('py1').className = 'tabbed-editor__node';

document.getElementById('java1').className = 'tabbed-editor__node';

document.getElementById('c1').className = 'tabbed-editor__node tabbed-editor__node--active';

document.getElementById('cpp1').className = 'tabbed-editor__node';

document.getElementById('python-code').className = 'code-editor__area';

document.getElementById('java-code').className = 'code-editor__area';

document.getElementById('c-code').className = 'code-editor__area code-editor__area--active';

document.getElementById('cpp-code').className = 'code-editor__area';

}

</script>

// <![CDATA[ <-- For SVG support

if ('WebSocket' in window) {

(function () {

function refreshCSS() {

var sheets = [].slice.call(document.getElementsByTagName("link"));

var head = document.getElementsByTagName("head")[0];

for (var i = 0; i < sheets.length; ++i) {

var elem = sheets[i];

var parent = elem.parentElement || head;

parent.removeChild(elem);

var rel = elem.rel;

if (elem.href && typeof rel != "string" || rel.length == 0 || rel.toLowerCase() == "stylesheet") {

var url = elem.href.replace(/(&|\?)_cacheOverride=\d+/, '');

elem.href = url + (url.indexOf('?') >= 0 ? '&' : '?') + '_cacheOverride=' + (new Date().valueOf());

}

parent.appendChild(elem);

}

var protocol = window.location.protocol === 'http:' ? 'ws://' : 'wss://';

var address = protocol + window.location.host + window.location.pathname + '/ws';

var socket = new WebSocket(address);

socket.onmessage = function (msg) {

if (msg.data == 'reload') window.location.reload();

else if (msg.data == 'refreshcss') refreshCSS();

};

if (sessionStorage && !sessionStorage.getItem('IsThisFirstTime_Log_From_LiveServer')) {

console.log('Live reload enabled.');

sessionStorage.setItem('IsThisFirstTime_Log_From_LiveServer', true);

}

})();

}

else {

console.error('Upgrade your browser. This Browser is NOT supported WebSocket for Live-Reloading.');

}

// ]]>

</script>

</body>

</html>

Provide feedback

Saved searches

Use saved searches to filter your results more quickly

FilesExpand file tree

dynamic-programming.html

Latest commit

History

dynamic-programming.html

File metadata and controls