Introduction to Algorithms (4th Edition, CLRS)
Authored by Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. Mapped to 6.006 assignment problems by our tutors.
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
MIT 6.006 Algorithms Homework Help
Problem set walkthroughs in Python with CLRS-aligned proofs for asymptotic analysis, hashing, BSTs, graphs, shortest paths, and dynamic programming. The hardest 6.006 grading deduction is on PS6 single-source shortest paths, where students implement Bellman-Ford without the early-termination check and the autograder times out at 60 seconds on the 10,000-vertex test case, the exact failure mode our tutors annotate inline. Verified CS graduates from MIT, Stanford, and CMU, starting at $20 per problem set, 12-hour average turnaround.
8 Recurring Assignments Covered with autograder pass guarantees
3 Required Textbooks Mapped to assignment problems
3 Languages Covered Course-relevant language coverage
12h Avg Turnaround Standard delivery; rush 4-6h available
Course Overview
About 6.006 at Massachusetts Institute of Technology
MIT 6.006 introduces algorithms across 13 weeks with 26 lectures, 13 recitations, and 7 problem sets. The Spring 2020 redesign by Erik Demaine, Jason Ku, and Justin Solomon restructured the course around 5 modules: (1) introduction and document distance, (2) data structures (sequence and set abstract data types, dynamic arrays, hashing, AVL trees, sorting), (3) graphs (BFS, DFS, topological sort, weighted shortest paths), (4) shortest paths (Dijkstra, Bellman-Ford, Johnson), and (5) dynamic programming (single subproblem, multiple subproblems, subset-sum, longest common subsequence). Lectures Monday and Wednesday at 11 AM in 32-123 (Ray and Maria Stata Center), recitations Friday at 11 AM.
Problem sets release every 2 weeks on Wednesday at 8 PM Eastern, due 13 days later Tuesday at 11 PM Eastern. Each problem set contains 4 to 6 problems mixing written analysis (asymptotic bounds, correctness proofs) and Python implementation (with PyPy 3 as the grading interpreter, allowing 2x to 10x speedup over CPython for algorithm-heavy workloads). Grading: 7 problem sets at 10 percent each (total 70 percent), 2 mid-term quizzes at 10 percent each, final exam at 10 percent.
The 6.006 lecture videos are mirrored at MIT OpenCourseWare and the textbook is CLRS 4th edition (Cormen, Leiserson, Rivest, Stein).
Course Reading
Required Textbooks for 6.006
6.006 Lecture Notes and Recitation Problems
Authored by Erik Demaine, Jason Ku, and Justin Solomon (MIT, 2020 redesign). Mapped to 6.006 assignment problems by our tutors.
Algorithm Design
Authored by Jon Kleinberg and Eva Tardos. Mapped to 6.006 assignment problems by our tutors.
Assignments
Recurring 6.006 Assignment Types
PS1 (Document Distance and Asymptotic Analysis)
Compute the document-distance angle between 2 text files via word-frequency vectors. Written analysis section asks for the asymptotic complexity of word_frequencies_for_file (O(n log n) for the sort-and-count version, O(n) for the hash-table version), with a 4-line proof. Expected runtime on the staff PyPy 3 grader: under 2 seconds for a 50,000-word file. The first pset establishes the problem-set format used through PS7.
PS2 (Sorting and Recurrences)
Implement merge sort and analyze its T(n) = 2 T(n/2) + Theta(n) recurrence via the master theorem, then solve a 3-part variant where the recurrence becomes T(n) = 2 T(n/2) + Theta(n log n) and the master theorem case 2 applies with k = 1. Coding section: implement an in-place 3-way quicksort with median-of-three pivot. Expected runtime on the grader: under 1 second for a 100,000-element random integer array, under 3 seconds for a 100,000-element sorted array.
PS3 (Hashing and BSTs)
Build a hash table with chaining and benchmark it against the staff reference implementation across 4 load factors (0.5, 1.0, 2.0, 5.0). Written analysis: prove the expected O(1) lookup cost under simple uniform hashing. Coding section: implement an AVL tree with insert, delete, and find supporting Theta(log n) for each operation. Grader timeout per test: 5 seconds on the 10,000-operation randomized sequence.
PS4 (Graph Traversal: BFS, DFS, Topological Sort)
Implement BFS and DFS on an adjacency-list graph representation with explicit color (white, gray, black) state per vertex. Solve the 3-part puzzle: (a) is there a path from s to t (BFS), (b) is the graph acyclic (DFS with cycle detection), (c) produce a topological ordering (DFS finish-time reverse). Written analysis: prove that DFS produces a valid topological ordering for any DAG via the lemma that descendants finish before ancestors.
PS5 (Dijkstra and Heap-Based Priority Queue)
Implement Dijkstra single-source shortest paths on a weighted directed graph with non-negative edge weights, using a binary-heap priority queue with decrease-key. Expected runtime on the grader: under 4 seconds on the 5,000-vertex 50,000-edge sparse graph test, under 8 seconds on the 1,000-vertex 200,000-edge dense graph. Written analysis: derive the O((V + E) log V) bound for the binary-heap version and the O(E + V log V) bound for the Fibonacci-heap version.
PS6 (Bellman-Ford and Currency Arbitrage)
Implement Bellman-Ford for single-source shortest paths with negative edge weights, including a negative-cycle detection pass after V minus 1 relaxation rounds. Application problem: detect a profitable currency-exchange cycle (e.g. USD to EUR to GBP to USD) by transforming exchange rates via negative log and running Bellman-Ford. Expected runtime: under 60 seconds on the 10,000-vertex test, with early termination when no relaxations occur in a round.
PS7 (Dynamic Programming: LCS and Knapsack)
Solve longest common subsequence (LCS) with the 2D recurrence dp[i][j] = dp[i-1][j-1] + 1 if a[i] equals b[j], else max(dp[i-1][j], dp[i][j-1]), then reconstruct the LCS via backward trace. Then 0/1 knapsack with weight and value vectors and capacity W. Written analysis: justify the choice of bottom-up tabulation over top-down memoization for the knapsack autograder (memory locality, no recursion overhead on the 1,000-item test).
Quiz 1 and Quiz 2 (60-minute closed-book midterms)
Quiz 1 (week 6) covers PS1 through PS3 material: asymptotic analysis, sorting recurrences, hashing, AVL trees. Quiz 2 (week 11) covers PS4 through PS6 material: BFS, DFS, topological sort, Dijkstra, Bellman-Ford. Format: 4 to 6 problems, 60 minutes, closed-book except a single-side 8.5x11 cheat sheet for each quiz. The final exam (3 hours, week 14) covers all material with a single 2-sided cheat sheet.
Common Pitfalls
Where 6.006 Students Get Stuck
PS1 word_frequencies hashing collision storm
The naive word_frequencies implementation uses a Python dict with default string hashing. For inputs constructed by an adversary (e.g. all words that hash to the same bucket under CPython 3.4 string hashing), the dict degenerates to O(n squared) and the grader times out at 60 seconds. The fix: PyPy 3 (which uses a randomized hash with per-process seed) plus a fallback to sorted-tuple counting if the dict size exceeds 100,000 entries.
PS2 master theorem case-2 boundary confusion
The recurrence T(n) = 2 T(n/2) + Theta(n log n) sits exactly on the boundary between cases 2 and 3 of the master theorem. n^log_2(2) equals n, and f(n) equals n log n which is Theta(n log^1 n). Case 2 with k = 1 applies, giving T(n) = Theta(n log^2 n). Students who guess case 3 (incorrectly thinking f(n) dominates) write T(n) = Theta(n log n) and lose 8 points on the rubric. The fix: explicitly compute log_b(a) and identify the k value before picking the case.
PS3 AVL delete rebalancing chain
AVL delete requires rebalancing from the deleted node's parent up to the root. Students who rebalance only at the deletion point miss the case where a rotation at depth d unbalances depth d - 1 by 1 (left-left or right-right become out-of-balance). The fix: a recursive rebalance helper that returns the new subtree root and is called along the path from the deleted node to the root, rotating at each level where the balance factor exceeds 1.
PS5 Dijkstra decrease-key on Python heapq
Python's heapq module does not support decrease-key. Students who implement Dijkstra with `heapq.heappush(pq, (new_dist, vertex))` and ignore stale entries see the autograder pass on small inputs and TLE on the 5,000-vertex test because the heap grows to E entries instead of V. The fix: lazy deletion (skip an entry if its distance is greater than the recorded best distance for that vertex when popped), which is the standard idiom in 6.006 staff solutions. Reduces effective heap size and matches the O((V + E) log V) bound.
PS6 Bellman-Ford missing early termination
Standard Bellman-Ford runs V minus 1 rounds of relaxation. The early-termination optimization: if a full round produces zero relaxations, the algorithm has converged and the remaining rounds are wasted. Without it, the 10,000-vertex test case runs all 9,999 rounds and times out at the 60-second grader budget. The fix: track a `relaxed` flag per round and break out of the outer loop when no edge relaxes.
PS6 negative cycle detection ambiguity
After V minus 1 rounds, any edge that still relaxes lies on (or is reachable from) a negative-weight cycle. Students who report the first such edge as "the negative cycle" miss credit because the cycle may be 5 or 10 edges long and starts elsewhere. The fix: starting from the edge that relaxed in round V, walk back V steps along the parent pointers to guarantee landing on a vertex inside the cycle, then walk forward following parents until returning to the start vertex.
PS7 LCS reconstruction off-by-one
The LCS dp table is 1-indexed in CLRS (rows 0 to m, columns 0 to n, with row 0 and column 0 as base cases). The reconstruction walks from dp[m][n] back to dp[0][0] following the recurrence. Students who 0-index in Python (rows 0 to m-1) reconstruct an LCS of length dp[m-1][n-1] but miss the final character. The fix: pad the input strings with a sentinel character at index 0 and use 1-indexed dp throughout.
PyPy 3 vs CPython benchmark divergence
6.006 grades on PyPy 3, which JIT-compiles hot loops and runs 2x to 10x faster than CPython 3 for algorithm-heavy code. Students who profile on CPython see runtime numbers that pass the grader by a comfortable margin, then submit and TLE because the grader actually uses CPython 3.10 on the older PS1 through PS3 graders. The fix: check the per-pset grader documentation at 6006.scripts.mit.edu for the interpreter version, and profile locally with the matching version.
Sample Work
6.006 Code from Past Deliveries
Every 6.006 deliverable ships with annotated code, an autograder transcript, and a line-by-line walkthrough. Browse anonymized samples to see what a delivered pset looks like before you submit.
See sample 6.006-style assignments we have delivered
Sample-work archive includes code, comments, autograder output, and the design-decision notes our tutors leave for each pset.
Browse sample workRelated Coverage
What Pairs With 6.006
Language
Python Homework Help
Annotated Jupyter notebooks and pytest-passing scripts for ML, pandas, and algorithm assignments, with PEP 8 formatting and type hints throughout. The most common failure in Data Science labs (Berkeley DATA 100, U of T STA130, Edinburgh INFR11125, NUS DSA1101, IIT Bombay DS203) and Intro Programming psets (Berkeley CS61A, U of T CSC108, Manchester COMP16321, Sydney COMP1531, NUS CS1010E) is silent NumPy broadcasting that produces the wrong output shape without raising, the exact failure mode our tutors catch with assert statements inline. Verified CS graduates from Georgia Tech, BITS Pilani, U of Toronto, Manchester, NUS, and IIT, starting at $20 per task, 12-hour average turnaround.
Subject
Algorithms Homework Help
Sorting, dynamic programming, greedy, divide-and-conquer, and graph algorithms with formal complexity analysis. The hardest CS161 grading deduction is failing to prove the greedy-choice property for a problem that looks greedy but is not, the gap our tutors close with a worked exchange argument. Verified CS graduates from Georgia Tech, Purdue, and BITS Pilani, starting at $20 per task, 12-hour average turnaround.
Subject
Data Structures Homework Help
Implementation, Big-O analysis, and JUnit edge-case tests for every structure from arrays through graphs. The number-one cause of failed Gradescope submissions in CS61B is iterator invalidation after a mid-traversal mutation, the exact failure mode our tutors annotate inline. Verified CS graduates from BITS Pilani, EPFL Lausanne, and Georgia Tech, starting at $20 per task, 12-hour average turnaround.
FAQ
6.006 Tutoring, Frequently Asked Questions
Do you help with all 7 MIT 6.006 problem sets?
Yes. All 7 problem sets from PS1 document distance through PS7 dynamic programming, plus the 2 quizzes and the final exam. Coverage spans the 5 course modules: introduction, data structures (sequence and set ADTs, sorting, hashing, AVL), graphs (BFS, DFS, topological sort), shortest paths (Dijkstra, Bellman-Ford, Johnson), and dynamic programming (LCS, knapsack, subset-sum). All Python deliverables target PyPy 3 with documented runtime expectations matching the staff grader budget for each pset.
How do you handle Dijkstra without decrease-key in Python?
Python heapq does not support decrease-key, so Dijkstra implementations use lazy deletion: on heappop, check whether the popped distance equals the recorded best distance for the vertex; if not, skip the entry. This keeps the heap size at O(E) instead of O(V) but does not change the asymptotic O((V + E) log V) bound because each edge generates at most one heap insert. The pattern matches the 6.006 staff solution from Spring 2020 onward.
Can you guarantee the PS6 Bellman-Ford under 60 seconds?
Standard PS6 Bellman-Ford deliveries complete the 10,000-vertex test in 8 to 15 seconds on PyPy 3 with early termination. The implementation tracks a `relaxed` flag per round and breaks the outer loop when no edge relaxes, typically converging in 50 to 200 rounds rather than the full V minus 1 = 9,999. Negative-cycle detection runs one additional round after convergence; if any edge still relaxes, the parent-pointer walkback recovers the cycle in O(V) extra time.
Do you cover the Demaine, Ku, Solomon 2020 redesign vs older 6.006?
Yes. The Spring 2020 redesign restructured the course around 5 modules and shifted problem set 4 from a complete-graph traversal pset to a BFS-DFS-topo-sort pset, and added the currency-arbitrage application to PS6. Pre-2020 problem sets (from the Erik Demaine and Srini Devadas era) used different inputs and slightly different grader budgets. Specify the semester (e.g. Spring 2024 vs Spring 2018) when submitting so the deliverable targets the matching staff solution and rubric.
Is using CSHH for MIT 6.006 allowed under the collaboration policy?
6.006 publishes its collaboration policy at courses.csail.mit.edu/6.006/spring24/collab.html: discussion with classmates is encouraged, but submitted writeups and code must be written individually. CSHH operates as a study reference: every deliverable includes a 5-line proof sketch in a code comment for each algorithm, asymptotic complexity bounds with derivations, and a recommendation to retype the solution after reading rather than copy-paste. Whether a specific submission complies with your section's interpretation of the policy is your judgment to make against the published rules.
Can you help with the 6.006 quizzes and final exam?
Yes. Live tutoring at $40 per hour for quiz prep (Quiz 1 covers PS1 through PS3 material in 60 minutes, Quiz 2 covers PS4 through PS6 material in 60 minutes) and final exam prep (3 hours, all material). Past exams from Spring 2017 through Spring 2024 are archived at 6006.scripts.mit.edu/spring24 and we work through every problem with the published solutions. Closed-book except a single-side 8.5x11 cheat sheet per quiz and a 2-sided cheat sheet for the final.
What about MIT 6.046 (advanced algorithms) vs 6.006?
6.046 is the graduate-level follow-on covering randomized algorithms, network flow, linear programming, and approximation, taught by Erik Demaine, Costis Daskalakis, and Adam Klivans. 6.006 is the undergraduate intro at the level of CLRS chapters 1 through 25. We cover both: 6.006 at $20 to $30 per pset, 6.046 at $35 to $50 per pset given the proof depth. Specify the course number on submission.
How do you handle the PS3 AVL tree delete tests?
AVL delete requires rebalancing from the deleted node's parent up to the root, with the recursive rebalance helper returning the new subtree root at each level. Our implementation uses the textbook 4-case rotation (LL, LR, RL, RR) with a unified rotate_left and rotate_right helper, and the balance check fires at every depth on the deletion path. The grader runs 10,000 randomized insert-delete sequences under @timeout(5) and our deliveries complete in 2 to 3 seconds on PyPy 3.
What turnaround do you offer on 6.006 problem sets?
12-hour average for PS1, PS2, PS4. 18 to 24 hours for PS3, PS5, PS7 given the implementation surface. PS6 currency arbitrage runs 24 to 36 hours given the proof + implementation combination. Pricing: $20 Debug and Explain per pset, $30 Full Solution per pset, $40 per hour Live Tutoring. Rush 4 to 6 hours available on PS1 and PS2 for an additional fee.
Do you help with 6.006 lecture videos and recitation problems?
Yes. The 26 lecture videos at MIT OpenCourseWare and the 13 recitation problem packets at 6006.scripts.mit.edu are standard study material. Live tutoring sessions walk through recitation problems with the same depth as office hours: we trace each example, prove correctness on the spot, and answer follow-up questions. The OCW Spring 2020 series (Demaine, Ku, Solomon) is the most current and matches the current pset structure.
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