Data Structures and Algorithms: cram pack

Formula sheets, the key method steps for every topic and 6 practice problems, with full solutions at the back.

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1Formulas and methods

Graph theory

Handshake lemma
Σ deg(v) = 2|E|
Complete graphs
|E(Kn)| = n(n − 1)/2, |E(Km,n)| = m·n
Regular graph
|E| = n·k/2
Tree and forest
|E| = n − 1, |E| = n − c
Leaves of a tree
L = 2 + Σ (deg(v) − 2) over deg(v) ≥ 3
Euler
circuit: all degrees even; path: exactly 2 odd degrees
Hamilton circuits in Kn
(n − 1)!/2
Dijkstra update
d(v) = min(d(v), d(u) + w(u, v))
Walks of length L
(Mᴸ)ij

Method

  1. Write down the degree of every vertex; the degrees add up to twice the number of edges.
  2. For Euler questions, check that the graph is connected and count the vertices with odd degree.
  3. For trees, use n − 1 edges and the handshake lemma.
  4. For Dijkstra, keep a table of tentative distances and settle the smallest unsettled vertex each round.
  5. For a minimum spanning tree, sort the edges by weight and skip every edge that closes a cycle (Kruskal), or grow one tree from a start vertex (Prim).

2Practice problems

1.Algorithms & complexityMedium

A recurrence T(n) = 2T(n/2) + O(n) solves to:

  1. AO(log n)
  2. BO(n²)
  3. CO(n log n)
  4. DO(n)
2.Algorithms & complexityHard

Which of these satisfiability problems is known to be solvable in polynomial time?

  1. ACircuit satisfiability
  2. BGeneral SAT on formulas in conjunctive normal form
  3. C3-SAT, where each clause has exactly three literals
  4. D2-SAT, where each clause has at most two literals
3.Data structuresMedium

An in-order traversal of a binary search tree visits the keys in:

  1. AAscending sorted order
  2. BDescending order always
  3. CInsertion order
  4. DLevel order
4.Data structuresHard

In a hash table using open addressing, why is deletion often done with a "tombstone" marker?

  1. ATombstones speed up hashing
  2. BEmptying the slot could break the probe sequence for other keys
  3. CIt makes the table sorted
  4. DDeleted keys must be kept forever for security, so slots can never be reused
5.Graph theoryMedium
ABCDEFEuler circuitEuler path, no circuitneither

Does the connected graph have an Euler circuit, an Euler path that is not a circuit, or neither? Click your answer.

|V| = 6|E| = 8
  1. 1Euler circuit
  2. 2Euler path, no circuit
  3. 3neither
6.Graph theoryHard
1263142191117719ABCDEFG

Find the total weight W of a minimum spanning tree of the weighted graph.

|V| = 7|E| = 11
W =

3Solutions

1.Algorithms & complexity

Answer: C · O(n log n)

Why this is the answer

By the master theorem (a = 2, b = 2, f(n) = n), T(n) = Θ(n log n).

Answer

C · O(n log n)
2.Algorithms & complexity

Answer: D · 2-SAT, where each clause has at most two literals

Why this is the answer

2-SAT is in P (linear time via strongly connected components), while 3-SAT is NP-complete.

Answer

D · 2-SAT, where each clause has at most two literals
3.Data structures

Answer: A · Ascending sorted order

Why this is the answer

Visiting the left subtree, then the node, then the right subtree outputs BST keys in sorted order.

Answer

A · Ascending sorted order
4.Data structures

Answer: B · Emptying the slot could break the probe sequence for other keys

Why this is the answer

If a deleted slot became empty, later searches might stop early and miss keys that were placed further along the probe chain.

Answer

B · Emptying the slot could break the probe sequence for other keys
5.Graph theory

Answer: neither

Euler, Hamilton and trees: a connected graph has an Euler circuit (every edge once, back to the start) when every vertex has even degree, and an Euler path that is not a circuit when exactly two vertices have odd degree; the path then runs from one odd vertex to the other. A Hamilton circuit visits every vertex once; Kn has (n − 1)!/2 of them. A tree on n vertices has n − 1 edges, and a connected graph with m edges keeps a spanning tree after removing m − n + 1 edges.

Degrees

deg(A) = 1, deg(B) = 3, deg(C) = 3, deg(D) = 3, deg(E) = 4, deg(F) = 2

Odd vertices

A, B, C, D → 4
neither

Answer

neither
6.Graph theory

Answer: W = 30

Shortest paths and spanning trees: Dijkstra’s algorithm settles one vertex per round: the unsettled vertex with the smallest tentative distance. Its distance is then final, and every neighbour v gets d(v) = min(d(v), d(u) + w(u, v)). Kruskal’s algorithm builds a minimum spanning tree by taking the edges from light to heavy and skipping any edge that would close a cycle; Prim’s algorithm grows one tree from a start vertex by always adding the lightest edge that leaves it. A spanning tree of n vertices has n − 1 edges.

Kruskal’s algorithm

DF (1): take
DE (2): take
BC (3): take
AC (6): take
GD (7): take
EF (9): skip, closes a cycle
GB (11): take

Total weight

W = 1 + 2 + 3 + 6 + 7 + 11 = 30

Answer

W = 30

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