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

Why does dynamic programming give no real speed-up for merge sort?

  1. AMerge sort is already O(n)
  2. BMerge sort has no recursive structure
  3. CDynamic programming only works on graph problems, not on arrays
  4. DIts subproblems do not overlap, so there is nothing to reuse
2.Algorithms & complexityHard

Which bound is correct for log(n!)?

  1. AΘ(n²)
  2. BΘ(n log n)
  3. CΘ(n)
  4. DΘ(log n)
3.Data structuresMedium

Which collision resolution method stores colliding keys in a list at the same bucket?

  1. AQuadratic probing
  2. BSeparate chaining
  3. CDouble hashing
  4. DLinear probing
4.Data structuresHard

Deleting a node with two children from a binary search tree is usually done by:

  1. ASwapping it with the root
  2. BReplacing it with its in-order successor (or predecessor) and deleting that node
  3. CRebuilding the subtree below it from scratch and reinserting every key it contained
  4. DRemoving the whole subtree
5.Graph theoryMedium

A connected graph has 17 vertices and 28 edges. How many edges k must be removed to leave a spanning tree?

|V| = 17|E| = 28
k =
6.Graph theoryHard
91551013121421118ABCDEFG

Prim’s algorithm builds a minimum spanning tree starting from vertex G. Click the edge it adds as edge number 2.

start: Gk = 2
  1. 1edge AB
  2. 2edge AC
  3. 3edge BC
  4. 4edge BD
  5. 5edge DE
  6. 6edge DF
  7. 7edge EF
  8. 8edge GB
  9. 9edge GC
  10. 10edge GD

3Solutions

1.Algorithms & complexity

Answer: D · Its subproblems do not overlap, so there is nothing to reuse

Why this is the answer

Merge sort's halves are disjoint and each is solved once; DP helps only when subproblems recur, as in Fibonacci.

Answer

D · Its subproblems do not overlap, so there is nothing to reuse
2.Algorithms & complexity

Answer: B · Θ(n log n)

Why this is the answer

log(n!) ≤ n log n, and the largest n / 2 factors give log(n!) ≥ (n / 2) log(n / 2); this is why comparison sorting needs Ω(n log n).

Answer

B · Θ(n log n)
3.Data structures

Answer: B · Separate chaining

Why this is the answer

Chaining keeps a linked list (or other structure) per bucket; open addressing looks for another free slot in the table itself.

Answer

B · Separate chaining
4.Data structures

Answer: B · Replacing it with its in-order successor (or predecessor) and deleting that node

Why this is the answer

The in-order successor has at most one child, so it can be removed easily after copying its key into the node being deleted.

Answer

B · Replacing it with its in-order successor (or predecessor) and deleting that node
5.Graph theory

Answer: k = 12

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

17 − 1 = 16

Edges to remove

k = 28 − 16 = 12

Answer

k = 12
6.Graph theory

Answer: edge BC

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.

Prim’s algorithm

1. GB (2)
2. BC (5)
3. AB (9)
4. BD (10)
5. DF (12)
6. DE (13)

edge number 2

BC

Answer

edge BC

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