, b, d, D), dimension of an OSN gives smallest number of attributes needed to identify users that is, given the graph structure, we can (theoretically) recover the social **space** **Complex** Networks100 6 Dimensions of Separation OSNDimension YouTube6 Twitter4 Flickr4 Cyworld7 **Complex** Networks101 Future directions what precisely is a community in an OSN? could help us with applications such as targeted advertising and counterterrorism/

-hard by showing that another NP-hard problem is polynomial- time reducible to it. (This establishes a lower bound on the **complexity** of the problem.) **Complexity** classes A **complexity** class is a class of problems grouped together according to their time and/or **space** **complexity** NC: can be solved very efficiently in parallel P: solvable by a DTM in poly-time (can be solved efficiently/

the maximum number of tape cells that M scans on any input of length n If the **space** **complexity** of M is f(n), we say that M runs in **space** f(n) **Space** **Complexity** accept Computation tree of a DTM …... f(n) : max number of used cells q/Theorem, PSPACE = NPSPACE, because the square of a polynomial function is still a polynomial. Outline Chapter 8 ： Definition of **Space** **Complexity** Savitch’s Theorem The Class PSPACE PSPACE-Completeness TQBF Problem PSPACE-Completeness Definition 8.8 A language B is PSPACE-complete if/

the “depth” parameter. This is the bottom-line irreducible worst case cost of systematic searches. 131SEARCH Worst Case **Complexity** 4 The number of states in the search **space** may be exponential in some “depth” parameter, e.g. number of actions in a plan, number of /of what states have been visited (expanded), searches can do (much) worse than visit every state. 132SEARCH Worst Case **Complexity** 4 N NA state **space** with N states may give rise to a search tree that has a number of nodes that is exponential in N,/

other” The result: multiple normative frameworks and operative mechanisms (overlapping, competing, complementing) Mapping the regime **complex** of **space** warfare Mapping: normative framework: principles, norms, rules, decision-making procedures (all ”norms”) Noting:/UNGA resolutions & Sino-Russian draft treaty on prevention of weaponization. Mechanisms = COPUOS, ITU Mapping the regime **complex** of **space** warfare Theater of cyberspace: norms = UN Group of Governmental Experts (GGE) – report + recommendations of /

N, where f(n) is the maximum number of steps that M uses on any branch of its computation on any input of length n. The **space** **complexity** of M is the function f: N N, where f(n) is the maximum number of tape cells that M scans on any branch of / g(n) = O(f (n)), if there exist constant n 0 and c such that for all n n 0, g(n) cf(n) **Space** **complexity** classes can be defined similarly. 2004 SDU 15 Relationship among models Theorem. For every t(n) n, each t(n) time multi-tape Turing machine has an /

that has exactly one or no satisfying assignment? Structural **Complexity** Classes P, NP, NP-completeness. **Space** bounded computation. Structural **Complexity** Classes P, NP, NP-completeness. **Space** bounded computation. How much **space** is required to check s-t connectivity? Structural **Complexity** Classes P, NP, NP-completeness. **Space** bounded computation. Counting **complexity**. Structural **Complexity** Classes P, NP, NP-completeness. **Space** bounded computation. Counting **complexity**. How hard is it to count the number of/

2009 Result #3: Successful greedy paths are shortest Regardless the structure of the hidden **space**, **complex** network topologies are such, that all successful greedy paths are asymptotically shortest But: how many greedy paths are successful /geometry of trees; the volume of balls grows exponentially with their radii Greedy routing in **complex** networks, including the real AS Internet, embedded in hyperbolic **spaces**, is always successful and always follows shortest paths Even if some links are removed, emulating/

. The developmental regime is associated with changes in the system. These two regimes cannot be sufficiently isolated for a **complex**-system. n Identify or define targeted outcome **spaces**: Outcome **spaces** are large sets of possible partial outcomes at specific scales and in specific regimes. The **complex**-system itself will choose the exact combinations of partial outcomes that it realizes. n Establish rewards (and penalties/

them unintentionally However, not enough to name and describe the concepts Can use the logic of **complexity** to: a. Understand the problem **space** when addressing apparently intractable problems b. Create enabling environments How & Whom? Policy makers who become/ workshop with all interviewers and sponsors 2-day Workshop to Identify the Problem **Space** 72 themes grouped into 8 clusters: OBU/CFBU Interface **Complexity** of structure (matrix) Human behaviours Cultures* Communication Leadership/role of central team/

Ohm’s & 2 Kirhoff’s. All laws A=B+C, A=B×C , A-1=B-1+C-1, have identical geometric interpretation! 13 true, 14 false facts; simple P-**space**, **complex** neurodynamics. Intuitive reasoning 5 laws are simultaneously fulfilled, all have the same representation: Question: If R2=+, R1=0 and V =0, what can be said about I, V1, V2 ? Find/

: There are n nodes in the list All data references are null Number of references in the list and **space** required: Required **space** Total Reference SizeOfSinglyLinkedListElementReference 1 head tail n*SizeOfSinglyLinkedListElementReference n next n*SizeOfObjectReference data Total **space** = (n + 2)*SizeOfSinglyLinkedListElementReference + n*SizeOfObjectReference Hence **space** **complexity** is O(n) List Creation An empty list is created as follows: Once created, elements can be inserted into/

© 2004 The MITRE Corporation. All rights reserved Engineering a **Complex** System: The Air & **Space** Operations Center (AOC) as a **Complex** Systems Exemplar Doug Norman Senior Technical Advisor, AOC-WS Dept Head, AF Battle Management / and – respond to conditions as they emerge n Guides garden into the desired outcome **space** Sounds like a **complexity** problem? © 2004 The MITRE Corporation. All rights reserved SDM 2004 What is **Complexity**? n A measure of potentiality n It does not mean Difficult to understand n Contrast/

purpose usually does not have subparts e.g. a drum, a spinning toy, a windup toy = one play **space** (when calculating the **complexity** of the play environment) Shipley, D. (1993). Empowering children: Play-based curriculum for lifelong learning. Scarborough, Ont. : Nelson Canada / acceptable for the age group: 1-2 years = 5-6 play **space**? 3-4 years = 3-4 play **space**? 5-6 years = 2-3 play **space**? Is there a need for: More simple units? More **complex** units? More super units? Is there a need to combine units? /

other types –often able to trade **SPACE** for TIME. –Faster solution that uses more **space** –Slower solution that uses less **space** CS 314Efficiency - **Complexity** 30 Big O **Space** Big O could be used to specify how much **space** is needed for a particular algorithm –/if willing to take longer –truly beautiful solutions take less time and **space** The biggest difference between time and **space** is that you cant reuse time. - Merrick Furst CS 314Efficiency - **Complexity** 31 Quantifiers on Big O It is often useful to discuss/

and of the three families of particles were defined. It was shown that the electromagnetic fields are caused by the effects of the **complex** **space** and that the model is compatible with the Maxwells equations This is the version v2.12 published 14.4.2014. email: virtanen./expanding speed or the cosmic multiplier a is less than R / and spirals cut each other in point C. Thus both **complex** **spaces** seem to have there a common point. The light trace at the spiral causes such an effect that the points of the curve/

store only those. Hard to do well! Memory becomes a bottleneck. Question: Is this inherent? Or can the right heuristics avoid the memory bottleneck? Proof **Complexity** & Sat Solvers Proof Size ≤ Time for Ideal SAT Solver Proof **Space** ≤ Memory for Ideal SAT Solver Many explicit hard UNSAT examples known with exponential lower bounds for Resolution Proof Size. Question: Is this also true for/

of CANYIELD(c start, c accept, 2 df(n) ). ” Proof of Savitch ’ s theorem (iii) 8.1.d It remains to analyze the **space** **complexity** of M. Whenever CANYIELD invokes itself recursively, it stores the current stage number and the values of c 1,c 2 and p on a stack so/for M ’ to work properly) will only take O(log f(n)) **space**. And, while trying an i, M ’ uses no more additional **space** than M does. This **space** can be recyc- led on every iteration, so the **space** **complexity** of M ’ remains O(f 2 (n)). PSPACE defined 8.2.a /

store only those. Hard to do well! Memory becomes a bottleneck. Question: Is this inherent? Or can the right heuristics avoid the memory bottleneck? Proof **Complexity** & Sat Solvers Proof Size ≤ Time for Ideal SAT Solver Proof **Space** ≤ Memory for Ideal SAT Solver Many explicit hard UNSAT examples known with exponential lower bounds for Resolution Proof Size. Question: Is this also true for/

k ---> ’ ’ = 3 Peter van Emde Boas: Games and **Complexity** Guangzhou 2009 Constant Factor Speed-up This yields automatic constant factor speed-up in **space**: **Space**( S(n) ) = **Space**( S(n)/k ) Snags: Input is not compressed! This may require additional steps and/diagonalization is possible Peter van Emde Boas: Games and **Complexity** Guangzhou 2009 **SPACE** COMPRESSION Downward Diagonalization If S 1 (n) > log(n) is **space** constructible and S 2 (n) = o(S 1 (n)) then **Space**( S 2 (n) ) **Space**( S 1 (n) ) On input i #/

to Understand & Design **Complex** Human Systems? The Multi-Agent-based Simulation Path R. H. Weber, Sr. P.E. The Aerospace Corporation (310) 336-5715 System & Operations Engineering Reference Model Military Utility **Space**-Based KEW * S/, innovation, adhocracy, organized “stovepipes” US neglect of intellectual infrastructure Maladaptive effects of Cold War on military-industrial **complex** Impact Loss of US industrial/economic competitiveness at macro-system level EU Airbus Consortium (Catia) & Japan Toyota/

shortest path, … Recap: **Complexity** Hierarchy Easy Hard PH EXP #P-complete/hard: #SAT, sampling, probabilistic inference, … 5 What is PSPACE? P-**SPACE**: “Polynomial **space**” as opposed to polynomial time **space**: amount of “working memory” / “notepad **space**” that an algorithm has at/On the road to a whole new range of applications: Strategic decision making Performance guarantees in **complex** multi-agent scenarios Secure communication and data networks in hostile environments Robust logistics planning in adversarial /

leaf at the RHS so DFS will expand all nodes (m is cutoff) =1 + b + b2+ ......... + b^m = O (b^m) **Space** **Complexity** how many nodes can be in the queue (worst-case)? at depth l < d we have b-1 nodes at depth d we have b nodes /d=5 d=cut-off DFS = 1+10+100,…,=111,111 IDS = 123,456 Ratio is Comments on Iterative Deepening Search **Complexity** **Space** **complexity** = O(bd) (since its like depth first search run different times) Time **Complexity** 1 + (1+b) + (1 +b+b2) + .......(1 +b+....bd) = O(bd) (i.e., asymptotically/

bytes to represent the trace back matrix June 24, 2005Space Efficient Alignment Algorithms6 Simple Improvement for Scoring Matrix In reality, the **space** **complexity** of the scoring matrix is only linear, i.e., O(2*min(m,n)) = O(min(m,n)) O/(n,m)) – if m < n, switch the sequences (or save a row of s and s reverse instead) Linear **space** **complexity**!! June 24, 2005Space Efficient Alignment Algorithms10 Project Teams and Presentation Assignments (Revised) Base Project (Global Alignment): Miguel and Joseph Extension /

contingency Double contingency System identity System identity Identity violation and adaptation Identity violation and adaptation **Complexity** **Complexity** 3 Communication systems Communication system Communication units 4 Communication Sender unit: Signals generated Receiver unit/ rules: conditional probability distributions over the **space** of possible communications, which are part of the system Grammatical rules: conditional probability distributions over the **space** of possible communications, which are part /

: two possible approaches 18 Approach A: Estimator-correlator Parameter estimator Parameter estimator Parameter estimator Estimator Metric Function Metric Function Metric Function Correlator arg max Obviously exponential **complexity** w.r.t. N 19 Approach B: Parameter **space** scan Known Parameter Detector Known Parameter Detector Known Parameter Detector Known Parameter Detector arg max Metric Function Metric Function Metric Function Metric Function Unfortunately, this method/

K F H C B A L G J D F C B G D E treewidth = 2 cycle cutset = 5 Spring 2007 ICS-275 40 Time-**Space** **complexity** of of w-cutset **Space**: O(exp(w)) W -cutset: a set that when removed the induced- width is w. c(w): size of w-cutset. Time: O(exp(w/AO Search) 1.43.6 (Caching on Cutset) + (AO Search) 1.63.4 (Caching on Cutset) + (BE) 0.75.3 Spring 2007 ICS-275 73 Time-**Space** **complexity** of of w-cutset **Space**: O(exp(w)) W -cutset: a set that when removed the induced- width is w. c(w): size of w-cutset. m(w): depth of AO w-/

nodes With such test Depth-first, backtracking and iterative deepening search loose their linear worst- case **space** **complexity**, for any guarantee to avoid all loops may require keeping an exponential number of expanded nodes in / search algorithms do not scale up, neither theoretically (exponential worst case time or **space** **complexity**) nor empirically (experimentally measured average case time or **space** **complexity**) Heuristic search algorithms do scale up to very large problem instances, in some /

outputs exactly f(n) symbols on input 1 n, and runs in time O(f(n) + n) and **space** O(f(n)). April 3, 201520 Proper **Complexity** Functions includes all reasonable functions we will work with –log n, √n, n 2, 2 n, n!, /201521 Hierarchy Theorems Does genuinely more **space** permit us to decide new languages? Theorem (**Space** Hierarchy Theorem): For every proper **complexity** function f(n) ≥ log n: **SPACE**(f(n)) ( **SPACE**(f(n) log f(n)). Proof: same ideas. April 3, 201522 Robust Time and **Space** Classes What is meant by “robust/

of characters that are acceptable totalDigits Specifies the exact number of digits allowed. Must be greater than zero whiteSpace Specifies how white **space** (line feeds, tabs, **spaces**, and carriage returns) is handled 33 XML Data Types (cont.) 34 Elements – **Complex** (cont.) What is a **Complex** Element? –A **complex** element is an XML element that contains other elements and/or attributes. –There are four kinds of/

A=B+C, A=B×C, A 1 =B 1 +C 1, have identical geometric interpretation! 13 True, 14 False facts; simple P-**space**, **complex** neurodynamics. Geometric representation of facts: + increasing, 0 constant, - decreasing. Ohm’s law V=I×R; Kirhoff’s V=V 1 +V 2. True (I/, A=B×C, A 1 =B 1 +C 1, have identical geometric interpretation! 13 True, 14 False facts; simple P-**space**, **complex** neurodynamics. Question in qualitative physics: if R 2 increases, R 1 and V t are constant, what happens with current and V 1, V 2/

), if elim-cond(b) is applied along ordering d when Y is a b-cutset then the **space** **complexity** of elim-cond(b) is O(n exp(b)), and its time **complexity** is O(n exp (|Y|+b)). Fall 2003 ICS 275A - Constraint Networks 19 Finding a b/For b=1, hybrid(1,1) is the non-separable components utilizing the cycle-cutset in each component. The **space** **complexity** of this algorithm is linear but its time **complexity** can be much better than the cycle-cutsets cheme or the non-separable component approach alone. Fall 2003 ICS 275A /

all n –there exists a TM M that outputs exactly f(n) symbols on input 1 n, and runs in time O(f(n) + n) and **space** O(f(n)). April 1, 2004CS151 Lecture 220 Proper **Complexity** Functions includes all reasonable functions we will work with –log n, √n, n 2, 2 n, n!, … –if f and g are proper then f/

Morris, 2008) When searching for a solution to its problems, life apparently does not traverse the entire combinatorial **space** of possibilities, but continues to discover the same solutions which suggest that optimality criteria (or variational principles) are at/, intelligence and anticipation What else is needed, apart from variation and selection? 28 Two types of **complexity** **Complexity** 1 29 Simple vs. **complex** 1 systems Single cause and single effect (one-to-one connection) Small changes in the cause/

rigor to IA research Usability, experimentation, ethnography Dealing with ambiguity and **complexity** is also intuitive Advanced Information Architecture- Fall 02 It focuses on digital (web-based) information **spaces** A set of items held by an information system and the relations /http://www.ils.unc.edu/gbnewby/papers/building4.html http://www.ils.unc.edu/gbnewby/papers/building4.html A **complex** information **space** (C) stores a total number (N) of information units in a medium (M) of storage A user (X) /

types of technological systems that are progressively able to do more for us, in a more networked and resilient fashion, using less resources (matter, energy, **space**, time, human and economic capital) to deliver any fixed amount of **complexity**, productivity, or capability. We are faced daily with many possible evolutionary choices in which to invest our precious time, energy, and resources, but only/

involvement of additional resources and means into the research process directly during spaceflight is provided Crew member’s role in **space** programs implementation 9 Some examples of crewmembers’ function uniqueness aboard human **space** **complexes** Mir ISS Aboard Mir orbiting **complex** 5 research facilities of Priroda hardware **complex** were repaired, aboard the ISS RS – the Laser communication system, instruments for crew Earth observation, etc.; during the/

wind turbine sited on flat surface ) 0.901.910.67 2.13 2.130.91 SUMMARY Factors affecting the **complex** dynamics of the wind farms were investigated in detail; Factors affecting the **complex** dynamics of the wind farms were investigated in detail; Turbine **spacing** Turbine **spacing** Wind farm layout (aligned and staggered) Wind farm layout (aligned and staggered) Upstream turbine operating (yaw) conditions Upstream/

How many object are in the scene and how they are distributed How **complex** each particular object is –Check out gametutorials Frustum culling tutorial Octrees Octrees are a **space** partitioning data structure –We saw Octrees before in the context of modeling / m is the number of triangles per object –So it becomes infeasible as scene **complexity** or object **complexity** increases Collision Detection The solution is two-fold: –**Space** Partitioning Reduces the n 2 term –Bounding Volumes Reduces the m 2 term Collision /

” and “Spectr-R” spacecrafts models, production of models of the new designed onboard instruments and units of the spacecrafts. Development of the work documentation on the equipment of Ground Control **Complex** and Ground **Complex** on the Data Receiving, Processing and Distribution of the **Space** **Complexes** “Electro-L” and “Spectr-R” Update of the existing onboard and ground hardware and software for the “Arctica/

of Competence by NN and LP Best Classifier for Benchmarking Data All Rights Reserved © Alcatel-Lucent 2008 22 Regions in **complexity** **space** where the best classifier is (nn,lp, or odt) vs. an ensemble technique Boundary-NonLinNN IntraInter-Pretop MaxEff-/-crossing edges randomly [Macia et al. 2008] or, create partitions with increasing resolution can create continuous cover of **complexity** **space** but, are the data similar to those arising from reality? All Rights Reserved © Alcatel-Lucent 2008 29 Ways/

Fields GCD Powers mod p Fermat, Roots of Unity, & Generators Z mod p vs **Complex** Numbers Cryptography Other Finite Fields Vector **Spaces** Colour Error Correcting Codes Linear Transformations Integrating Changing Basis Fourier Transformation (sine) Fourier Transformation /b a×b=1, i.e. b=a -1 Examples: Reals & Rationals **Complex** Numbers Integers Invertible Matrices ( & a×0 = 0) Fields Problems for computers: Reals Too much **space** Lack of precision Integers Lack of inverses Grow too big Better field? Finite field, /

one can maintain a “current pointer” and do searches both forwards and backwards along the list. Disadvantages: extra **complexity** in the data structures; more **space** used, especially if the Items are small. int Delete(ItemType *X, ListNode **L, int pos) { / Master Pointers...... Heap After Compaction 91.102 - Computing II All of these methods are **complex**, and all of them involve various time-**space**-**complexity** trade-offs. Unfortunately, the moment we moved away from the single-task personal computer (once/

a 2D computational grid... Favorable property: better exploitation of the 2D locality due to the recursive nature / self-similarity. Application of **space**-filling curves JASS 2005 Saint Petersburg Application of **space**-filling curves 1. Representation of computational grids (1) Acceptable computational **complexity** is required in implemen- ting computational grids. Especially for adaptively refined grids the manipulation part cannot be too expensive choice of/

of any cylinder). Input: S = {S 1, …, S n } a collection of n simply-shaped bodies in d -**space** of constant description **complexity**. The problem: What is the maximal number of vertices/edges/faces that form the boundary of the union of the bodies in S/{F F} F(x), for x R 2. The **complexity** envelopes [Sharir 1994] The combinatorial **complexity** of the lower envelope of n simple algebraic surfaces in d -**space** is O*(n d-1 ). For d=3, the **complexity** of the lower envelope: O*(n 2 ) The sandwich region [ Agarwal/

/gnclab/Conference.html http://www.nps.edu/academics/gnclab/Conference.html 18 Thank you for your time! Jesus Isarraras isarrara@usc.edu 19 BACKUP CHARTS 20 CONCEPT - **COMPLEX** SUBSYSTEMS Large **Space** Aperture Architecture Comparison ALDCSTHSTJWST Herschel **Space** Observatory Type of MirrorSegmentedMonolithicSegmentedMonolithic Primary Aperture (m) 202.4 / 0.36.53.5 Mirror Mass (kg) 635 (mirrors, actuators) 828705300 (full telescope) Wavelength ( μm).11 - 2/

-optimal step –Reduce probability(non-optimal) over time Comparison to Hill Climbing –Completeness? –Speed? –**Space** **Complexity**? temp Limited Discrepancy Search Discrepancy bound indicates how often to violate heuristic Iteratively increase... a b c/ search with admissible heuristic –Plus keep checking until all possibilities look worse Evaluation –Finds optimal solution? –Time **Complexity**? –**Space** **Complexity**? Yes O(b^d) Underestimates cost of any solution which can reached from node Admissable Heuristics f(x)/

on in the system. The most remarkable feature to be stressed in the sudden transition from simple to **complex** behavior is the order and coherence of this system. This suggest the existence of correlations that is statistically. Long Range Correlation The characteristic **space** dimension of Benard cell in usual laboratory conditions is in the order 10 -1 cm the whereas the characteristic/

bugs, ensure compatibility across different versions Maintenance 5 3. Algorithm Analysis **Space** **complexity** How much **space** is required Time **complexity** How much time does it take to run the algorithm 6 **Space** **Complexity** **Space** **complexity** = The amount of memory required by an algorithm to run to completion/ e.g. actual text - load 2GB of text VS. load 1MB of text 8 Time **Complexity** Often more important than **space** **complexity** **space** available tends to be larger and larger time is still a problem for all of us 3-4GHz/

nodes With such test Depth-first, backtracking and iterative deepening search loose their linear worst- case **space** **complexity**, for any guarantee to avoid all loops may require keeping an exponential number of expanded nodes in / search algorithms do not scale up, neither theoretically (exponential worst case time or **space** **complexity**) nor empirically (experimentally measured average case time or **space** **complexity**) Heuristic search algorithms do scale up to very large problem instances, in some /

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