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

In a sorted array, you look for two elements that sum to a target by moving one pointer from each end inwards. What is the time complexity?

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

Which two things must be shown to prove that a decision problem X is NP-complete?

  1. AX has no polynomial algorithm, and X is decidable
  2. BX is in P, and X reduces to SAT
  3. CX reduces to a problem in P, and X is in NP
  4. DX is in NP, and a known NP-complete problem reduces to X in polynomial time
3.Data structuresMedium

What is the cost of inserting an element in the middle of an array of n elements?

  1. AO(n), because later elements must shift
  2. BO(1), because the index is known
  3. CO(log n), because of binary search
  4. DO(n²), because each element moves n times
4.Data structuresHard

An AVL node is unbalanced because its left child's right subtree became too tall (left-right case). Which fix is applied?

  1. AA single right rotation at the node
  2. BA single left rotation at the node
  3. CA right rotation at the left child, then a left rotation at the node
  4. DA left rotation at the left child, then a right rotation at the node
5.Graph theoryMedium

A connected graph has 11 vertices and 17 edges. How many edges k must be removed to leave a spanning tree?

|V| = 11|E| = 17
k =
6.Graph theoryHard
162121117187ABCDE

Kruskal’s algorithm builds a minimum spanning tree. Click the edge it adds as edge number 3; skipped edges do not count.

k = 3
  1. 1edge AB
  2. 2edge AC
  3. 3edge BC
  4. 4edge BD
  5. 5edge CE
  6. 6edge DE
  7. 7edge BE

3Solutions

1.Algorithms & complexity

Answer: B · O(n)

Why this is the answer

If the sum is too small the left pointer moves right, if too large the right pointer moves left, so there are at most n steps.

Answer

B · O(n)
2.Algorithms & complexity

Answer: D · X is in NP, and a known NP-complete problem reduces to X in polynomial time

Why this is the answer

Membership in NP shows X is not harder than NP; the reduction from a known NP-complete problem shows X is at least as hard as all of NP.

Answer

D · X is in NP, and a known NP-complete problem reduces to X in polynomial time
3.Data structures

Answer: A · O(n), because later elements must shift

Why this is the answer

All elements after the insertion point move one position to the right, which takes time proportional to n.

Answer

A · O(n), because later elements must shift
4.Data structures

Answer: D · A left rotation at the left child, then a right rotation at the node

Why this is the answer

The first rotation converts the left-right case into a left-left case, which a right rotation at the unbalanced node then fixes.

Answer

D · A left rotation at the left child, then a right rotation at the node
5.Graph theory

Answer: k = 7

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.

Edges of a spanning tree

11 − 1 = 10

Edges to remove

k = 17 − 10 = 7

Answer

k = 7
6.Graph theory

Answer: edge BD

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

1. AC (2)
2. BE (7)
3. BD (11)
4. BC (12)

edge number 3

BD

Answer

edge BD

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