Architecture networks and complexity John Doyle John G
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Architecture, networks, and complexity John Doyle John G Braun Professor Control and dynamical systems Bio. Engineering, Electrical Engineering Caltech
NRC theory report: Bad news and good news Bad: Attempts to connect with theory • Topology, modularity, information, … Good: Biology motivation • Diversity • Metabolism • Cell interior • Architecture (is not topology) • Robustness • Decision • Behavior
Alternative: Essential ideas • Listening to physicians, biologists, and engineers • Robust yet fragile (RYF) • “Constraints that deconstrain” (G&K) • Unity creating diversity • • Network architecture Layering Control and dynamics (C&D) Hourglasses and Bowties
Collaborators and contributors (partial list) Biology: Csete, Yi, El-Samad, Khammash, Tanaka, Arkin, Savageau, Simon, Af. CS, Kurata, Smolke, Gross, Kitano, Hucka, Sauro, Finney, Bolouri, Gillespie, Petzold, F Doyle, Stelling, Caporale, … Theory: Parrilo, Carlson, Murray, Vinnicombe, Paganini, Mitra Papachristodoulou, Prajna, Goncalves, Fazel, Liu, Lall, D’Andrea, Jadbabaie, Dahleh, Martins, Recht, many more current and former students, … Web/Internet: Li, Alderson, Chen, Low, Willinger, Kelly, Zhu, Yu, Wang, Chandy, … Turbulence: Bamieh, Bobba, Mc. Keown, Gharib, Marsden, … Physics: Sandberg, Mabuchi, Doherty, Barahona, Reynolds, Disturbance ecology: Moritz, Carlson, … Finance: Martinez, Primbs, Yamada, Giannelli, … Current Caltech Former Caltech Longterm Visitor Other
Thanks to you for inviting me, and • • • NSF ITR AFOSR NIH/NIGMS ARO/ICB DARPA Lee Center for Advanced Networking (Caltech) Boeing Pfizer Hiroaki Kitano (ERATO) Braun family
My interests Multiscale Physics Core theory challenges Sustainability? Systems Biology & Medicine
Bacterial networks • Necessity in chemotaxis • Design principles in heat shock response • Architecture of metabolism • Origin of high variability and power laws • Architecture of the cell • Control of core metabolism and glycolytic oscillations SBML/SBW SOSTOOLS Wildfire ecology Physiology and medicine (new) Publications: Science, Nature, Cell, PNAS, PLOS, Bioinfo. , Trends, IEEE Proc. , IET Sys. Bio, FEBS, PRL, … Systems Biology & Medicine
Wilbur Wright on Control, 1901 • “We know how to construct airplanes. ” (lift and drag) • “Men also know how to build engines. ” (propulsion) • “Inability to balance and steer still confronts students of the flying problem. ” (control) • “When this one feature has been worked out, the age of flying will have arrived, for all other difficulties are of minor importance. ”
Feathers and flapping? Or lift, drag, propulsion, and control?
Recommendations • (Obviously…) More and better theory • Need an “architecture” for research that is as networked as biology and our best technologies • Create the right “waist” of the research hourglass • E. g. find the constraints that deconstrain
Human complexity Robust J Efficient, flexible metabolism J Complex development and J Immune systems J Regeneration & renewal 4 Complex societies ÿ Advanced technologies Yet Fragile L Obesity and diabetes L Rich microbe ecosystem L Inflammation, Auto-Im. L Cancer N Epidemics, war, … M Catastrophic failures • Evolved mechanisms for robustness allow for, even facilitate, novel, severe fragilities elsewhere • often involving hijacking/exploiting the same mechanism • There are hard constraints (i. e. theorems with proofs)
Peter Sterling and Allostasis food intake Blood Glucose Oxygen Amino acids Fatty acids Universal metabolic system Organs Tissues Cells Molecules
sports Universal reward systems music Prefrontal dance cortex crafts VTA dopamine art Accumbens toolmaking sex food Dopamine, Ghrelin, Leptin, …
sports Universal reward systems music Prefrontal dance cortex crafts VTA dopamine art Accumbens toolmaking sex food Blood Glucose Oxygen Universal metabolic system Organs Tissues Cells Molecules
Universal reward systems work family community nature food sex toolmaking sports music dance crafts art VTA dopamine Prefrontal cortex Accumbens Robust and adaptive, yet …
work family community nature sex food toolmaking sports music dance crafts art VTA dopamine Prefrontal cortex Accumbens
work family community nature market/ consumer culture money salt sugar/fat nicotine alcohol VTA dopamine Accumbens Vicarious sex food toolmaking sports music dance crafts art Prefrontal cortex industrial agriculture
work family community nature money salt sugar/fat nicotine alcohol VTA dopamine Accumbens Vicarious sex toolmaking sports music dance crafts art Prefrontal cortex cocaine amphetamine
high sodium hypertension obesity atherosclerosis overwork diabetes smoking inflammation money salt sugar/fat nicotine alcohol Vicarious alcoholism drug abuse immune suppression coronary, cerebrovascular, renovascular cancer cirrhosis accidents/ homicide/ suicide
hypertension high sodium money salt sugar/fat nicotine alcohol Vicarious VTA obesity atherosclerosis dopamine overwork diabetes smoking inflammation alcoholism Glucose immune suppression coronary, cerebrovascular, renovascular cancer cirrhosis Oxygen drug abuse accidents/ homicide/ suicide
Human complexity Robust J Efficient, flexible metabolism J Complex development and J Immune systems J Regeneration & renewal 4 Complex societies ÿ Advanced technologies Yet Fragile L Obesity and diabetes L Rich microbe ecosystem L Inflammation, Auto-Im. L Cancer N Epidemics, war, … M Catastrophic failures • Evolved mechanisms for robustness allow for, even facilitate, novel, severe fragilities elsewhere • often involving hijacking/exploiting the same mechanism • There are hard constraints (i. e. theorems with proofs)
Robust yet fragile Systems can have robustness of – Some properties to – Some perturbations in – Some components and/or environment Yet fragile to other properties or perturbations. Many issues are special cases, e. g. : • Efficiency: robustness to resource scarcity • Scalability: robustness to changes in scale • Evolvability: robustness of lineages on long times to possibly large perturbations
Case studies Today (primary): • Cell biology Today (secondary): • Internet • Toy example: Lego • Wildfire ecology • Physiology • Power grid • Manufacturing • Transportation Other possibilities: • Turbulence • Statistical mechanics • Physiology (e. g. HR variability, exercise and fatigue, trauma and intensive care) • RYF physio (e. g. diabetes, obesity, addiction, …) • Disasters statistics (earthquakes)
Bio and hi-tech nets Exhibit extremes of • Robust Yet Fragile • Simplicity and complexity • Unity and diversity • Evolvable and frozen • Constrained and deconstrained What makes this possible and/ or inevitable? Architecture
• We use this word all the time. • What do we really mean by it? • What would a theory look like? Architecture
Human complexity Robust J J 4 ÿ Efficient, flexible metabolism Complex development and Immune systems Regeneration & renewal Complex societies Advanced technologies Yet Fragile L L N M Obesity and diabetes Rich microbe ecosystem Inflammation, Auto-Im. Cancer Epidemics, war, … Catastrophic failures • It is much easier to create the robust features than to prevent the fragilities. • There are poorly understood “conservation laws” at work
Robust yet fragile Most essential challenge in technology, society, politics, ecosystems, medicine, etc: • Managing spiraling complexity/fragility • Not predicting what is likely or typical • But understanding what is catastrophic (though perhaps rare) What community will step up and be central in this challenge?
Perturbations Robust yet fragile Perturbations Systems requirements: functional, efficient, robust, evolvable, scalable System and architecture Components and materials
Architecture= Constraints System-level Aim: a universal Emergent taxonomy of complex systems and theories in a r t s n o c e d t a h t s t n i a r t Component Con Protocols • Describe systems/components in terms of constraints on what is possible • Decompose constraints into component, systemlevel, protocols, and emergent • Not necessarily unique, but hopefully illuminating nonetheless
fan-in of diverse inputs Diverse function Universal Control Diverse components universal carriers fan-out of diverse outputs Universal architectures • Hourglasses for layering of control • Bowties for flows within layers
Evolution of theory • • • Verbal arguments (stories, cartoons, diagrams) Data and statistics (plots, tables) Modeling and simulation (dynamics, numerics) Analysis (theorems, proofs) Synthesis (hard limits on the achievable, reverse engineering good designs, forward engineering new designs) All levels interact and iterate
Example: Theory of planetary motion • • • Verbal Data & stats Model & sim Analysis Synthesis (Ptolemy, Copernicus) (Brahe, Galileo, Kepler) (Newton, Einstein) (Lagrange, Hamilton, Poincare) (NASA/JPL) All levels interact and iterate
Drill down Verbal Data/stat Mod/sim Analysis Synthesis • • • Describe theory Show some math Just to give a flavor You can ignore details Always return to verbal descriptions and handwaving summaries
Synthesis theories: Limits and tradeoffs On systems and their components • Thermodynamics (Carnot) • Communications (Shannon) • Control (Bode) • Computation (Turing/Gödel) No networks Assume different architectures a priori.
Hard limits and tradeoffs On systems and their components • Thermodynamics (Carnot) • Communications (Shannon) • Control (Bode) • Computation (Turing/Gödel) • Fragmented and incompatible • Cannot be used as a basis for comparing architectures • New unifications are encouraging No dynamics or feedback
Hard limits and tradeoffs On systems and their components • Thermodynamics (Carnot) • Communications (Shannon) • Control (Bode) • Computation (Turing/Gödel) • Include dynamics and feedback • Extend to networks • New unifications are encouraging Robust/ fragile is unifying concept
Why glycolytic oscillations? • • • Various answers depend on meaning of “why” Will go deeper into “why” using stages… Start with simplest possible models Motivate generalizable and scalable methods Extremely familiar and “done” problem in biology and dynamics at the small circuit level • Convenient to introduce new theory and thinking using the most familiar possible examples
Basics of glyc-oscillations • Verbal arguments (stories, cartoons, diagrams) • Data and statistics (plots, tables) Result: Cells and extracts show oscillatory behavior. Why?
Why? Modeling and simulation • Verbal arguments (stories, cartoons, diagrams) • Data and statistics (plots, tables) • Modeling and simulation (dynamics, numerics) • Why = propose mechanism, model, simulate, compare with data • Has been done extensively for this problem • What’s new? Simplicity and robustness
consumption Autocatalytic x Control y reaction metabolite reaction
Precursors Core metabolism Carriers rs a g Su ds i c A o Amin Nucleotides Fatty aci ds Cofact ors
Precursors Catabolism Carriers
Gly G 1 P G 6 P F 1 -6 BP Catabolism Gly 3 p 13 BPG ATP 3 PG 2 PG NADH Oxa PEP Pyr ACA TCA Cit
Gly Precursors G 1 P G 6 P F 1 -6 BP Gly 3 p 13 BPG 3 PG 2 PG Oxa PEP Pyr ACA TCA Cit
Gly Precursors G 1 P G 6 P F 6 P Autocatalytic F 1 -6 BP Gly 3 p Carriers ATP 13 BPG 3 PG 2 PG NADH Oxa PEP Pyr ACA TCA Cit
Gly G 1 P G 6 P Regulatory F 6 P F 1 -6 BP Gly 3 p ATP 13 BPG 3 PG 2 PG NADH Oxa PEP Pyr ACA TCA Cit
Gly G 1 P G 6 P F 1 -6 BP Gly 3 p 13 BPG 3 PG 2 PG Oxa PEP Pyr ACA TCA Cit
If we drew the feedback loops the diagram would be unreadable. Gly G 1 P G 6 P F 1 -6 BP Gly 3 p ATP 13 BPG 3 PG 2 PG Oxa PEP Pyr ACA TCA NADH Cit
Stoichiometry or mass and energy balance Biology is not a graph. Interna l Nutrients Products
Stoichiometry plus regulation Matrix of integers “Simple, ” can be known exactly Amenable to high throughput assays and manipulation Bowtie architecture Vector of (complex? ) functions Difficult to determine and manipulate Effected by stochastics and spatial/mechanical structure Hourglass architecture Can be modeled by optimal controller (? !? )
Gly S G 1 P G 6 P F 1 -6 BP Gly 3 p ATP 13 BPG Stoichiometry matrix 3 PG 2 PG NADH Oxa PEP Pyr ACA TCA Cit
Gly G 1 P G 6 P F 1 -6 BP Gly 3 p Regulation of enzyme levels by transcription/translation/degradation 13 BPG 3 PG 2 PG Oxa PEP Pyr ACA TCA Cit
Gly G 1 P G 6 P F 1 -6 BP Gly 3 p ATP 13 BPG Allosteric regulation of enzymes 3 PG 2 PG NADH Oxa PEP Pyr ACA TCA Cit
Gly G 1 P G 6 P Allosteric regulation of enzymes F 6 P Regulation of enzyme levels F 1 -6 BP Gly 3 p ATP 13 BPG 3 PG 2 PG NADH Oxa PEP Pyr ACA TCA Cit
Fa st Gly G 1 P res po ns e G 6 P Allosteric regulation of enzymes F 6 P Regulation of enzyme levels F 1 -6 BP Gly 3 p ATP 13 BPG 3 PG 2 PG NADH Slo w Oxa PEP Pyr ACA TCA Cit
Gly G 1 P G 6 P F 1 -6 BP Gly 3 p ATP 13 BPG 3 PG 2 PG NADH Oxa PEP Pyr ACA TCA Cit
Autocatalytic x Control y F 6 P F 1 -6 BP Gly 3 p 13 BPG 3 PG ATP
Autocatalytic x Control y F 6 P F 1 -6 BP Gly 3 p 13 BPG 3 PG ATP
Autocatalytic x Control y F 6 P F 1 -6 BP Gly 3 p 13 BPG 3 PG ATP
Autocatalytic x y Autocatalytic Control
Autocatalytic x Control y
x y
x y
x error 1 0. 5 0 -0. 5 -1 10 15 time 20
1 x error 0. 5 V=1. 1 0 -0. 5 -1 0 V=3 V=10 5 10 15 time 20
Why? Modeling and simulation • Why = propose mechanism, model, simulate, compare with data • Scalable to larger systems? Yes • Nonlinear? Yes • Explore parameter space? Awkward • Explore sets of uncertain models? Awkward
Why: Analysis • • Verbal arguments (stories, cartoons, diagrams) Data and statistics (plots, tables) Modeling and simulation (dynamics, numerics) Analysis (theorems, proofs) • Why = parameter regimes of instability, global results with nonlinearities
slow • Explicit regions of (in)stability • Easy to compare with experiments • Oscillations caused by • nonzero q (autocatalytic) • small k (low enzyme) • large V (high flux) • large h (strong inhibition) • Slow response caused by • large q (autocatalytic) oscillations • small V (low flux) • small h (weak inhibition)
1 0. 5 0 -0. 5 -1 0 oscillations 5 10 15 20
1 0. 5 0 -0. 5 -1 0 5 10 15 20 Nonlinear 1. 5 1 0. 5 0 0 10 20 30 40 50 60
1 0. 5 0 -0. 5 -1 0 5 10 15 20
1 Conservation law? 0. 5 0 -0. 5 -1 0 5 10 15 20
Analysis issues • Why = parameter regimes of instability, global results with nonlinearities • • • Scalable to larger systems? Less than sim Nonlinear? Yes Explore parameter space? Better than sim Explore sets of uncertain models? Better than sim Prove what models can’t do? Yes • Major research frontier
Why: Synthesis • Are there intrinsic tradeoffs or is this a “frozen accident”? (The former. ) • What are the relevant engineering principles? • How to separate necessity from accident? • Are there hard limits or conservation laws that apply? (Yes) • Is biology near these limits? (Apparently) • Why does autocatalysis and other efficiency issues aggravate regulation? (Stay tuned)
1 x 0. 5 0 -0. 5 -1 0 5 10 time 15 20 3 2 1 0 Spectrum -1 -2 0 1 2 3 freq 4 5
1 x 0. 5 0 -0. 5 -1 0 5 10 Theorem: time 15 20 3 2 1 0 -1 -2 0 1 2 3 freq 4 5
1 x 0. 5 0 -0. 5 Theorem: -1 0 5 10 time 15 20 3 2 1 0 -1 -2 0 1 2 3 freq 4 5
Why: Synthesis • There are too many hard limits on achievable performance to show in one hour… • Most are aggravated by – large q (more autocatalysis) – small V and k (less enzyme) • Thus tradeoffs between control response and efficiency • Can summarize with hand-waving argument. • Why = it’s an inevitable consequence of engineering tradoffs.
x y
Why: Synthesis (to do) • • There are hard contraints and tradeoffs Biology is hard up against these limits Yet there remains “design freedom” Why these particular “choices”? • What has evolution optimized? • Robustness (and evolvability)?
[ATP] 1. 05 1 h=3 0. 95 Time response 0. 9 Yet fragile 0. 85 0. 8 h=0 0 5 10 Time (minutes) 15 20 0. 8 h=3 Robust Log(Sn/S 0) 0. 6 Spectrum 0. 4 0. 2 h=0 0 -0. 2 -0. 4 -0. 6 -0. 8 0 2 4 Frequency 6 8 10
Yet fragile 0. 8 h=3 Robust Log(Sn/S 0) 0. 6 0. 4 0. 2 h=0 0 -0. 2 -0. 4 -0. 6 -0. 8 0 2 4 Frequency 6 8 10
Yet fragile 0. 8 Robust Log(Sn/S 0) 0. 6 0. 4 0. 2 h=0 0 -0. 2 -0. 4 -0. 6 -0. 8 0 2 4 Frequency 6 8 10
Theorem Transients, Oscillations 0. 8 Log(Sn/S 0) Tighter steady-state regulation h=3 0. 6 0. 4 h=2 0. 2 h=0 0 h=1 -0. 2 -0. 4 -0. 6 -0. 8 0 2 4 Frequency 6 8 10
This tradeoff is a law. log|S | Transients, Oscillations Tighter regulation Biological complexity is dominated by the evolution of mechanisms to more finely tune this robustness/fragility tradeoff.
cost = amplification goal: make this small • benefits = attenuation of disturbance • goal: make this as negative as possible Constraint: benefits costs
Bode e=d-u -u d Plant Control • What helps or hurts this tradeoff? • Helps: advanced warning, remote sensing • Hurts: instability, remote control
e=d-u -u d Plant Control Freudenberg and Looze, 1984
Bode e=d-u -u d Plant Control benefits costs stabilize
Bode e=d-u -u d Plant Control benefits Negative is costs good stabilize
e=d-u Plant - d Disturbance Remote Sensor Control Channel Control Sensor Channel Encode Nuno C Martins and Munther A Dahleh, Feedback Control in the Presence of Noisy Channels: “Bode-Like” Fundamental Limitations of Performance. Nuno C. Martins, Munther A. Dahleh and John C. Doyle Fundamental Limitations of Disturbance Attenuation in the Presence of Side Information (Both in IEEE Transactions on Automatic Control) http: //www. glue. umd. edu/~nmartins/
Electric power network Variety of producers Variety of consumers • Good designs transform/manipulate energy • Subject to hard limits
Standard interfaces n i a r t s n o c e d t a h nt t i a r t s n Co Variety of producers • • • Energy carriers Variety of consumers 110 V, 60 Hz AC (230 V, 50 Hz AC) Gasoline ATP, glucose, etc Proton motive force
e=d-u Plant Disturbance d Control Channel Control Sensor Channel benefits Fragile Remote Sensor Encode stabilize Robust remote control feedback costs • • remote sensing Good designs transform/manipulate robustness Subject to hard limits Unifies theorems of Shannon and Bode (1940 s) Claim: This is the most crucial (known) limit against which network complexity must cope
Bode e=d-u -u d Plant Control benefits costs stabilize
Bode e=d-u -u d Plant Control benefits Negative is costs good stabilize
Bode’s integral formula Yet fragile Robust benefits costs
e=d-u - d Disturbance u Plant Cost of stabilization Control Cost of control benefits costs
e=d-u Plant Cost of remote control Channel Control benefits costs
e=d-u Plant - Disturbance d Control Channel Control benefits stabilize costs remote control feedback
e=d-u Plant - d Remote Sensor Control Channel Sensor Channel Control benefits Disturbance stabilize costs Encode remote control feedback remote sensing
e=d-u Plant - d Remote Sensor Control Channel Control Disturbance Sensor Channel Encode Benefit of remote sensing benefits costs
e=d-u Plant - d Remote Sensor Control Channel Sensor Channel Control benefits Disturbance stabilize costs Encode remote control feedback remote sensing
e=d-u Plant - d Remote Sensor Control Channel Control Disturbance Sensor Channel Encode Bode/Shannon is likely a better p-to-p comms theory to serve as a foundation for networks than either Bode or Shannon alone.
Electric power network Variety of producers Variety of consumers • Good designs transform/manipulate energy • Subject to hard limits
e=d-u Plant Disturbance d Control Channel Control Sensor Channel benefits Fragile Remote Sensor Encode stabilize Robust remote control feedback costs • • remote sensing Good designs transform/manipulate robustness Subject to hard limits Unifies theorems of Shannon and Bode (1940 s) Claim: This is the most crucial (known) limit against which network complexity must cope
[a system] can have [a property] robust for [a set of perturbations] Fragile Yet be fragile for [a different property] Or [a different perturbation] Robust
[a system] can have [a property] robust for [a set of perturbations] • Some fragilities are inevitable in robust complex systems. Fragile Robust • But if robustness/fragility are conserved, what does it mean for a system to be robust or fragile?
Emergent Fragile • Some fragilities are inevitable in robust complex systems. Robust • But if robustness/fragility are conserved, what does it mean for a system to be robust or fragile? • Robust systems systematically manage this tradeoff. • Fragile systems waste robustness.
Electric power network Variety of producers Variety of consumers • Good designs transform/manipulate energy • Subject (and close) to hard limits
e=d-u Plant Control Channel Control d Disturbance Fragile Remote Sensor Control Encode Channel Robust • Robust designs transform/manipulate robustness • Subject (and close) to hard limits • Fragile designs are far away from hard limits and waste robustness.
DNA replication Trans* Proteins Constraints Genes Hard constraints: Thermo (Carnot) Info (Shannon) Control (Bode) Compute (Turing) s gar u S cids Fatty a Co-factors Amin o Aci Nu ds cle o Carriers tid es Precursors Catabolism Systems requirements: functional, efficient, robust, evolvable Diverse Universal Control Diverse Components and materials: Energy, moieties Protocols
End of part 1 Questions?
- Space complexity of nested loops
- Datagram network diagram
- Robert venturi complexity and contradiction in architecture
- Basestore iptv
- Evelyn doyle ucd
- Short doyle
- Short doyle
- Shama joshi
- Mary conan doyle
- Wendy doyle
- West sussex ccg merger
- Mariane asad doyle
- Who created the fictional detective sherlock holmes
- Jeremiah doyle
- Nash v inman
- Dissociated meaning
- Sir arthur conan doyle contribution forensic science
- Doyle mcmanus political affiliation
- Doyle
- The red headed league theme
- 2016679
- Sir arthur conan doyle cae answers
- Celia doyle
- Belinda doyle
- Single node architecture in wireless sensor networks
- Bluetooth in computer networks
- Osi architecture in computer networks
- Call and return architecture in software engineering
- Uninformed search examples
- What is kaplan model
- Language ne demek
- Spin glasses and complexity
- Depth and complexity frames
- Details icon depth and complexity
- Linear search average performance
- Which nims management characteristics includes developing
- Kaplans icons
- Divide and conquer complexity
- Depth and complexity questions for reading
- Global oncology trends 2017 advances complexity and cost
- Entropy order parameters and complexity
- Cis 262
- Architecture business cycle activities
- Modular product architectures
- Integral architecture example
- Bus design in computer architecture
- Networks and graphs circuits paths and graph structures
- Advantages and disadvantages of wired and wireless networks
- Leq structure ap world
- Diff between primitive and nonprimitive
- Dfsdef
- A sorting technique is called stable if: *
- O(n^2) time complexity
- Recurrence relation merge sort
- Complexity of differential privacy
- V5-7gzofadq -site:youtube.com
- Shell sort complexity
- Difference between bubble sort and selection sort
- Time complexity of linear search
- Space complexity bfs
- Recursion time complexity
- Polylogarithmic time complexity
- Recursion time complexity
- Cyclomatic complexity example
- Matrix multiplication time complexity
- Merge sort complexity
- Recursion algorithm
- Depth first search algorithm complexity
- Cost complexity pruning
- Function synonym math
- Cognitive complexity theory hci
- Matrix multiplication time complexity
- Min heap insertion time complexity
- 2 power n time complexity
- Data coverage in software testing
- Cyclomatic complexity example
- Sample complexity for finite hypothesis spaces
- Diagonalization method in theory of computation
- Derivational theory of complexity
- Pre order traversal
- Asymptotic complexity analysis
- How to calculate time complexity in data structure
- Early adulthood cognitive development
- Dd path testing
- Multistage graph
- Integrating strategy and culture in internal assessment
- Wallow in complexity
- Modularity in vlsi design means
- Explicitness academic writing
- Time complexity of ternary search
- Depth first search algorithm complexity
- Amortized time big o
- Sifat sifat struktur data tree
- Time complexity of algorithms
- 2 power n time complexity
- Algorithm complexity
- Min heap insertion time complexity
- Polylogarithmic time complexity
- Bst worst case time complexity
- Complexity of nested for loops
- Essential complexity
- The computational complexity of linear optics
- Algorithm complexity exercises
- Cyclomatic complexity of binary search
- Vc complexity
- Bfs
- Amortized complexity
- Kruskal complexity
- Acknowledging complexity meaning
- Big o notation if else statement
- Algorithm complexity analysis
- Service complexity definition
- Strategic management internal assessment
- Exemplifies the complexity of relationships